Lectures on Disintegration of Measures

Lectures on
Disintegration of Measures
By
L. Schwartz
Tata Institute of Fundamental Research
Bombay
1976
Lectures on
Disintegration of Measures
By
L. Schwartz
Notes by
S. Ramaswamy
Tata Institute of Fundamental Research
Bombay
1975
c
Tata Institute of Fundamental Research, 1975
No part of this book may be reproduced in any
form by print, microfilm or any other means without written permission from the Tata Institute of
Fundamental Research, Colaba, Bombay 400005
Preface
THE CONTENTS OF these Notes have been given as a one month
course of lectures in the Tata Institute of Fundamental Research in
March 1974. The present reduction by Mr. S. Ramaswamy gives under a
very short form the main results of my paper “Surmartingales régulières
à valeurs mesures et désintégrations régulières d’une measure” which
appeared in the Journal d’Analyse Mathematique, Vol XXVI, 1973. For
a first reading these Lecture Notes are better than the complete paper
which however contains more results, in more general situations. Nothing is said here about stopping times. On the other hand, the integral
representation with extremal elements of §3 in Chapter VII here, had
not been published before.
L. Schwartz
v
Note
The material in these Notes has been divided into two parts. In part
I, disintegration of a measure with respect to a single σ-algebra has
been considered rather extensively and in part II, measure valued supermartingales and regular disintegration of a measure with respect to an
increasing right continuous family of σ-algebras have been considered.
The definition, remarks, lemmata, propositions, theorems and corollaries have been numbered in the same serial order. To each definition
(resp. remark, lemma, proposition, theorem, corollary) there corresponds a triplet (a, b, c) where ‘a’ stands for the chapter and ‘b’ the
section in which the definition (resp. remark, lemma, proposition, theorem, corollary) occurs and ‘c’ denotes its serial number. References to
the bibliography have been indicated in square brackets.
The inspiring lectures or Professor L. Schwartz and the many discussions I had with him, have made the task of writing these Notes easier.
S. Ramaswamy
vii
Contents
Preface
v
Note
vii
I Disintegration of a measure with respect to a single...
1 Conditional expectations and disintegrations
1
Notations . . . . . . . . . . . . . . . . . . . . . . .
2
Basic definitions in the theory of... . . . . . . . . . .
3
Conditional Expectations and Disintegrations;... . . .
4
Illustrations and motivations . . . . . . . . . . . . .
5
Existence and uniqueness theorems of... . . . . . . .
6
Existence and Uniqueness theorems of conditional...
7
Another way of proving the existence... . . . . . . .
8
A few properties of conditional... . . . . . . . . . . .
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3
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2 Measure valued Functions
21
1
Basic definitions: Fubini’s theorem... . . . . . . . . . . . 21
2
Fubini’s theorem for Banach space... . . . . . . . . . . . 24
3 Conditional expectations of measure valued...
1
Basic definition . . . . . . . . . . . . . .
2
Preliminaries . . . . . . . . . . . . . . .
3
Uniqueness Theorem . . . . . . . . . . .
4
Existence theorems . . . . . . . . . . . .
ix
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Contents
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6
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Existence theorem for disintegration... . . . . . . . . . . 56
Another kind of existence theorem... . . . . . . . . . . . 57
Conditions for a given family... . . . . . . . . . . . . . . 61
II Measure valued Supermartingales and...
69
4
Supermartingales
71
1
Extended real valued Supermartingales . . . . . . . . . . 71
2
Measure valued Supermartingales . . . . . . . . . . . . 76
3
Properties of regular measure valued supermartingales . 78
5
Existence and Uniqueness of...
1
Uniqueness theorem . . . . . . . . . . . . . . . . . . . .
2
Existence theorem . . . . . . . . . . . . . . . . . . . . .
3
The measures νtw and the left limits . . . . . . . . . . . .
6
Regular Disintegrations
97
1
Basic Definitions . . . . . . . . . . . . . . . . . . . . . 97
2
Properties of regular disintegrations . . . . . . . . . . . 98
7
Strong σ-Algebras
1
A few propositions . . . . . . . . . . .
2
Strong σ-Algebras . . . . . . . . . . .
3
Choquet-type integral representations .
4
The Usual σ-Algebras . . . . . . . . .
5
Further results on regular disintegrations
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111
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124
Part I
Disintegration of a measure
with respect to a single
σ-Algebra
1
Chapter 1
Conditional expectations and
disintegrations
1 Notations
Throughout these Notes the following notations will be followed. R will 1
stand for the set of all real numbers, R̄, for the set of all extended real
numbers, i.e. R together with the ideal points −∞ and +∞, N for the set
of all natural numbers, Z for the set of all integers and Q for the set of
all rational numbers. R+ (resp R̄+ ) will stand for the positive elements
of R (resp R̄).
Let X be a non-void set. Let A be a subset of X. Them ∁A will stand
for the complementary set of A, i.e. ∁A = {x ∈ X | x < A}.
Let X be a non-void set and X a σ-algebra of subsets of X. Then
the pair (X, X) is called a measurable space. If f is a function on a
measurable space (X, X ), with values in a topological space Y, Y , ∅,
we say f belongs to X and write f ∈ X if f is measurable with respect
to X. (For measurablility concepts, we always consider on topological
spaces only their Borel σ-algebra, i.e., the σ-algebra generated by all
the open sets).
By a measure space (X, X, µ) we always mean a measurable space
(X, X) and a positive, . 0 measure µ on X.
Let (X, X, µ) be a measure space. Let A be a subset of X. We say µ
3
4
1. Conditional expectations and disintegrations
is carried by A if µ(∁A) = 0 in case A ∈ X and in case A < X, if µ(B)
for every B ∈ X, B ⊂ ∁A is zero. A ⊂ X is said to be a µ-null set if there
exists a B ∈ X with µ(B) = 0 and A ⊂ B. Nµ will stand for the class of
2
all µ-null sets of X. X̂µ will stand for the σ-algebra generated by X and
Nµ .
X̂µ is called the completion of X with respect to µ. If X = X̂µ , we say
X is complete with respect to µ. If f is a function on X with values in a
topological space Y, Y , ∅ and if f ∈ X̂µ , we say f is µ-measurable. If
Y is any σ-algebra on X, we denote by Y VNµ , the σ-algebra generated
by Y and Nµ . Thus, X̂µ = XVNµ .
The symbol ∀µ x will stand for ‘for µ-almost all x’.
If h is any non-negative function ǫ X̂µ , h, µ will stand for the measure
R
on X̂µ given by h · µ(B) = h(w)dµ(w) for all B ∈ X̂µ .
B
If E is a Banach space over the real numbers, L1 (X; X; µ; E) will
stand for the Banach space of all µ-equivalence classes of functions on
X with values in E which belong to X̂µ and which are µ-integrable.
L1 (X; X; µ) will stand for the Banach space of all µ-equivalence
classes of functions with values in R̄ which belong to X̂µ and which
are µ-integrable.
Let A (X; X; µ) denote the set of all extended real valued functions
on X which belong to X̂µ and let A + (X; X; µ) denote the set of all nonnegative elements of A (X; X; µ).
l
If Rf ∈ A + (X; X; Rµ) or if f ∈ Ll (X;
R X; µ) of if f ∈ L (X; X; µ; E), then
µ( f ), f (w)dµ(w), f (w)µ(dw), f dµ will all denote the integral of
f with respect to µ, over X.
2 Basic definitions in the theory of integration for
Banach space valued functions
3
The theory of integration for Banach space valued functions on a measure space is assumed here. However, by way of recalling, we give
below a few basic definitions.
Let (Ω, O, λ) be a measure space. Let E be a Banach space over the
3. Conditional Expectations and Disintegrations;...
5
real numbers. If S is any σ-algebra on Ω, a function f on Ω with values
in E is said to be a step function belonging to S, if there exist finitely
many sets (Ai )i=1,2,...,n , Ai ∈ S, i = 1, . . . n, Ai ∩ A j = ∅ if i , j and
finitely many points (xi )i=1,...n , xi ∈ E ∀i = 1, . . . n such that ∀w ∈ Ω,
n
P
f (w) =
χAi (w)xi . A function f on Ω with values in E is said to be
i=1
strongly measurable if there exists a sequence ( fn )n∈N of step functions,
fn ∈ Ôλ ∀n ∈ N, such that ∀λ w, fn (w) → f (w) in E, as n → ∞. Note
that a strongly measurable function belongs to Ôλ . A function f on Ω
with values in E is said to be λ-integrable orR integrable in the sense of
Bochner if f is strongly measurable and if | f |(w)dλ(w) < ∞, where
| f | is the real valued function on Ω assigning to each w ∈ Ω, the norm
of f (w) in E. One can prove that if S is a σ-algebra contained in Ôλ
and if f ∈ S is λ-integrable, then one can find a sequence ( fn )n∈N of step
functions belonging to S and a real valued non-negative λ-integrable
function g belonging to S such that
∀λ w, fn (w) → f (w) in E as n → ∞ and
∀n, ∀α w, | fn |(w) ≤ g(w).
For further properties of Bochner integrals, the reader is referred to
Hille and Phillips [1].
3 Conditional Expectations and Disintegrations;
Basic definitions
Let (Ω, O, λ) be a measure space. Let C be a σ-algebra contained in Ôλ .
Definition 1. Let f be a λ-integrable function on Ω with values in a
Banach space E over the reals (resp. f extended real valued, f ≥ 0 and
f ∈ Ôλ ). A function f C on Ω with values in E (resp. extended reals) is 4
said to be a conditional expectation of f with respect to C if
C
(i) f C ∈ C and is Rλ-integrable (resp.
R (i) f ∈ C and is ≥ 0)
and (ii) ∀ A ∈ C , f C (w)dλ(w) = f (w)dλ(w).
A
A
1. Conditional expectations and disintegrations
6
Here,
R
A
f (w)dλ(w) (resp.
R
f C (w)dλ(w)) stands for the integral of
A
χA . f (resp. χA . f C ) with respect to λ. These exist since f and f C are
λ-integrable (resp. f and f C are ≥ 0).
Definition 2. A family (λCw )w∈Ω of positive measures on O, indexed by
Ω is called a system of conditional probabilities with respect to C or a
disintegration of λ with respect to C if it has the following properties,
namely
(i) ∀B ∈ O, the function w → λCw (B) belongs to C
and (ii) ∀B ∈ O, the function w → λCw (B) is a conditional expectation
of χB with respect to C .
We remark that (i) implies that for every function f on Ω, f ≥ 0,
f ∈ O, the function w → λCw ( f ) belongs to C and that (ii) implies that
∀ function f on Ω, f ≥ 0, f ∈ O, the function w → λCw ( f ) belongs to C
and that (ii) implies that ∀ function f on Ω, f ≥ 0, f ∈ O, the function
w → λCw ( f ) is a conditional expectation of f with respect to C . From
(ii), taking B = Ω, we see that ∀λ w, λCw is a probability measure.
Note that the existence of a disintegration of λ with respect to C implies immediately the existence of conditional expectations with respect
to C for non-negative functions belonging to O. We shall see below that
if λ restricted to C is σ-finite, conditional expectation with respect to C
for any non-negative function belonging to Ôλ and for any λ-integrable
Banach space valued function exists. But a disintegration of λ need not
always exist without any further assumptions about Ω and O as can be
seen by an example due to J. Dieudonné [1].
4 Illustrations and motivations
5
Let (Ω, O, λ) be a measure space. Before we proceed to prove the existence theorems of conditional expectations, we shall see the above notions when C is given by a partition of Ω.
χA · λ
Let A ∈ Ôλ be such that 0 < λ(A) < ∞. We call the measure
λ(A)
4. Illustrations and motivations
on O, the conditional probability of λ given A. We call
7
R
f dλ
A
, the
λ(A)
conditional expectation of f given A, where f is either a Banach space
valued λ-integrable function or a non-negative function belonging to Ôλ .
Note that the conditional probability is always a probability measure
whether λ is or not.
Suppose also that 0 < λ(∁A) < ∞. (This will happen only if λ
is a finite measure and when 0 < λ(A) < λ(Ω)). Then, consider the
σ-algebra C = (∅, A, ∁A, Ω).
Let E be a Banach space over the real numbers and let f be a λintegrable function on Ω with values in E (resp. f ≥ 0, f ∈ Ôλ ). Define
R
f dλ




A



 Rλ(A) , if w ∈ A
f C (W) = 

f dλ




∁A


, if w ∈ ∁A
λ(∁A)
Then f C is with values in E, f C ∈ C and is λ-integrable (resp. f C
is ≥ 0, f C ∈ C ).
We have
R
R
(i) f C (w)dλ(w) = f (w)dλ(w) and
A
(ii)
R
A
A
f C (w)dλ(w) =
R
f (w)dλ(w).
∁A
Thus, f C is a conditional expectation of f with respect to C .
Consider the family (λCw )w∈Ω of measures on O defined as
χ · λ
A



, if w ∈ A



λ(A)
C
λw = 
χ∁A · λ




, if w ∈ ∁A.

λ(∁A)
This family (λCw )w∈Ω satisfies the conditions (i) and (ii) of definition
(1, § 3, 2) and hence is a disintegration of λ with respect to C .
6
1. Conditional expectations and disintegrations
8
Thus, we see that the conditional expectation of any Banach space
valued λ-integrable function, that of any non-negative function belonging to Ôλ and a disintegration of λ with respect to C , always exist when
C is of the form (∅, A, ∁A, Ω) with 0 < λ(A) < ∞ and 0 < λ(∁A) < ∞.
Next, let us consider the following situation.
S
Suppose Ω =
An , where ∀n, An ∈ Ôλ , An ∩ Am = ∅, if n , m
n∈N
and 0 < λ(An ) < ∞ ∀n ∈ N. We call such a sequence (An )n∈N of sets,
a partition of Ω. Let I be any subset of N. Then, the collection of all
S
sets of the form Ai , as I varies over all the subsets of N is a σ-algebra.
i∈I
Let us denote this σ-algebra by C . We say then that C is given by the
partition (An )n∈N .
If f is a λ-integrable function with values in E or if f is ≥ 0 and
belongs to Ôλ , consider the function f C defined as
C
f (w) =
R
f dλ
An
λ(An )
, if w ∈ An .
Then f C is easily seen to be a conditional expectation of f with
respect to C .
Consider the family (λCw )w∈Ω of measures on O defined as
λCw =
χ An · λ
if w ∈ An .
λ(An )
7
Then the family (λCw )w∈Ω of measures satisfies the condition (i) and
(ii) of definition (1, § 3, 2) and hence is a disintegration of λ with respect
to C .
Thus, we see that when C is given by a partition (An )n∈N with 0 <
λ(An ) < ∞, conditional expectation of any Banach space valued λintegrable function, that of any non-negative function belonging to Ôλ
and a disintegration of λ with respect to C , always exist.
5. Existence and uniqueness theorems of...
9
5 Existence and uniqueness theorems of conditional
expectations for extended real valued functions;
A few properties
Let (Ω, O, λ) be a measure space. Let C be a σ-algebra contained in
Ôλ . Let further, λ restricted to C be σ-finite.
Proposition 3. Let f ∈ L1 (Ω; O; λ) (resp. f ∈ A + (Ω; O; λ)). Then
a conditional expectation of f with respect of C exists. Moreover, it is
unique in the sense that if g1 and g2 are two conditional expectations of
f with respect to C , then ∀λ w, g1 (w) = g2 (w).
Proof. The proof of this proposition follows just from a straight forward
application of the
R Radon-Nikodym theorem. More precisely, for A ∈ C ,
define ν(A) = f (w)dλ(w). Then ν is a finite signed measure (resp. a
A
positive measure) on C and is absolutely continuous with respect to λ
which is σ-finite on C . Hence, by the Radon-Nikodym theorem, there
exists a function g ∈ C which is λ-integrable (resp. g ∈ C and is ≥ 0)
and which is unique upto a set of measure zero, such that
Z
Z
ν(A) =
f (w)dλ(w) =
g(w)dλ(w) ∀A ∈ C .
A
A
Note that for any given f ∈ L1 (Ω; O; λ) (resp. f ∈ A + (Ω; O; λ), 8
we get a class of functions as conditional expectations of f with respect
to C , any two functions in the class differing by a C -set of measure
zero at most. Also note that if f1 and f2 are any two functions belonging to A + (Ω; O; λ) and are equal almost everywhere, then also we have
∀λ w, f1C (w) = f2C (w). Thus, we have a map uO,C (resp. vO,C ) from
L1 (Ω; O; λ) to L1 (Ω; O; λ) (resp. from A + (Ω; O; λ) to A˜C+ where Ã+C
stands for the set of all λ-equivalence classes of non-negative functions
which belong to C ). Here, we are making as abuse of notation by denoting by the same symbol f, a function as well as the class to which it
belongs.
1. Conditional expectations and disintegrations
10
The map uO,C is a continuous linear map from the
1
L (Ω; O; λ) to L1 (Ω; C ; λ) with ||uO,C || = 1 where ||uO,C ||
Banach
R space
= sup
f ,0
| f C |dλ
R
.
| f |dλ
We have the following properties of the maps uO,C and vO,C .
(i) uO,C ( f C ) = f C ∀ f ∈ L1 (Ω; O; λ) i.e. uO,C is a projection onto the
subspace L1 (Ω; C ; λ) of L1 (Ω; O; λ)
(vO,C ( f C ) = f C ∀ f ∈ A + (Ω; O; λ))
(ii) |uO,C ( f )| ≤ uO,C (| f |) in the sense that ∀ f ∈ L1 (Ω; O; λ), ∀λ w,
|uO,C ( f )|(w) ≤ uO,C (| f |)(w).
(iii) f ≥ 0 ⇒ uO,C ( f ) ≥ 0 in the sense that ∀ f ∈ L1 (Ω; θ; λ) which is
such that ∀λ w, f (w) ≥ 0, we have ∀λ w, uO,C ( f )(w) ≥ 0.
(iv) If g is non-negative and belongs to C and f ∈ L1 (Ω; O; λ) and if
g f is λ-integrable or if g is λ-integrable and belongs to C such
that g f is λ-integrable, we have
uO,C ( f g) = g · uO,C ( f ).
(If g is non-negative and belongs to C ) and if f ∈ A + (Ω; O; λ),
then
(∀λ w, vO,C ( f g)(w) = g(w) · vO,C ( f )(w)).
9
(v) If S is any σ-algebra contained in Cˆλ ,
uC ,S ◦ uO,C = uO,S
i.e. the operation of conditional expectation is transitive
(vC ,S ◦ vO,C = vO,S )
(vi)
uO,O = Identity.
(∀ f ∈ A + (Ω; O; λ),
(vii)
uO,(Ω,∅) ( f ) =
(∀ f
∈
1 R
f dλ if λ(Ω) is finite
λ(Ω)
A + (Ω; O; λ),
if λ(Ω) < ∞)
∀λ w, vO,O ( f )(w) = f (w))
∀λ w, vO,(Ω,∅) ( f )(w)
=
1 R
f dλ
λ(Ω)
6. Existence and Uniqueness theorems of conditional...
11
6 Existence and Uniqueness theorems of
conditional expectation for Banach space
valued integrable functions
Let (Ω, O, λ) be a measure space, and let C be a σ-algebra contained in
Ôλ . Let λ restricted to C be σ-finite. Let E be a Banach space over the
real numbers.
Now, we are going to prove the existence and uniqueness theorems
of conditional expectations for any f ∈ L1 (Ω; O; λ; E). To prove the
existence theorem, we have to adopt essentially a different method than
the one in § 5 for extended real numbers as the Radon-Nikodym theorem in general is not valid for Banach spaces. By the Radon-Nikodym 10
theorem for Banach spaces we mean the following theorem:
Let (X, X, µ) be a measure space. Let E be a Banach space over
the real numbers. Let ν be a measure on X with values in E and let ν be
absolutely continuous with respect to µ. Then, there exists a µ-integrable
function g on χ with values in E such that
Z
∀ A ∈ X, ν(A) =
gdµ.
A
Theorem 4. Let f ∈ L1 (Ω; O; λ; E). Then,
(i) Existence: A conditional expectation of f with respect to exists.
(ii) Uniqueness: If g1 and g2 are two conditional expectations of f
with respect to C , then ∀λ w, g1 (w) = g2 (w).
N
Proof.
(i) Existence. Let L1 (Ω; O, λ)
E be the algebraic tensor
R
product of L1 (Ω; O; λ) and E over the real
There is
N numbers.
an injective linear map from L1 (Ω; O; λ)
E to L1 (Ω; O; λ; E)
RN
which takes an element f ⊗ x of L1 (Ω; O; λ)
E to f.x of L1 (Ω;
R N
O; λ; E). Hence, we can consider L1 (Ω; O; λ)
E as a subR
space of L1 (Ω; O; λ; E). A theorem of Grothendieck says that
1. Conditional expectations and disintegrations
12
L1 (Ω; O; λ; E) is the completion of L1 (Ω; O; λ)
N
topology’ on L1 (Ω; O; λ)
E.
R
N
E for the ‘π-
R
N
N
E to L1 (Ω; C ; λ)
E
R
R
N
which takes an element f ⊗ x of L1 (Ω; O; λ)
E to uO,C ( f ) ⊗ x
R
N
of L1 (Ω; C ; λ)
E is a contraction mapping of these normed
The linear map v from L1 (Ω; O; λ)
R
spaces under the respective ‘π-topologies’ and hence extends to a
unique continuous linear mapping of L1 (Ω; O; λ; E) to L1 (Ω; C ;
λ; E), which we again denote by v, Now, it is easy to see that
∀ f ∈ L1 (Ω; O; λ; E), v( f ) is a conditional expectation of f with
respect to C .
11
(ii) Uniqueness. To prove uniqueness, it is sufficient to prove that if
f is a λ-integrable
function on Ω with values in E and belongs to
R
C and if f dλ = 0∀A ∈ C , then ∀λ w, f (w) = 0.
A
So, let f ∈ L1 (Ω; C ; λ; E) with
R
f dλ = 0 ∀A ∈ C .
A
′
Let E ′ be the topological dual of E. If x′ ∈ E ′ and x ∈ E,
R let hx , xi
denote the value of x′ at x, ∀x′ ∈ E ′ , ∀A ∈ C , we have hx′ , f (w)i
A
dλ(w) = 0, and therefore, ∀x′ ∈ E ′ , ∀λ w, hx′ , f (w)i = 0.
Now, there exists a set N1 ∈ C with λ(N1 ) = 0, and a separable
subspace F of E such that if w < N1 , f (w) ∈ F. This is because, f is the
limit almost everywhere of step functions. Since F is separable, there
exists a countable set (x′n )n∈N , x′n ∈ F ′ ∀n ∈ N, such that
∀x ∈ F, ||x|| = sup |hx′n , xi|
n
(See Hille and Phillips [1], p.34, theorem 2.8.5). Therefore, if w <
N1 ,
| f |(w) = sup |hx′n , f (w)i|
n
7. Another way of proving the existence...
13
where | f |(w) is the norm of the element f (w).
Since ∀x′ ∈ E ′ , ∀λ w, hx′ , f (w)i = 0, we can find a set N2 ∈ C with
λ(N2 ) = 0 such that if w < N2 , hx′n , f (w)i = 0 ∀n ∈ N.
Therefore, if w < N1 ∪ N2 ,
| f |(w) = sup |hx′n , f (w)i| = 0.
n
This shows that
∀λ w, f (w) = 0.
7 Another way of proving the existence theorem of
conditional expectations for Banach space valued
integrable functions
The existence of conditional expectation for Banach space valued inte- 12
grable functions can be proved in the following way also.
As before, let (Ω, O, λ) be a measure space. Let C be a σ-algebra
contained in Ôλ . Let λ restricted to C be σ-finite. Let E be a Banach
space over the real numbers. Let E ′ be the topological dual of E. If
x′ ∈ E ′ and x ∈ E, hx′ , xi will stand for the value of x′ at x. If f is a
function on Ω with values in E and if ξ is a function on Ω with values in
E ′ , hξ, f i will stand for the real valued function on Ω associating to each
w ∈ Ω, the real number hξ(w), f (w)i. If f and ξ are as above, | f | (resp.
|ξ|) will denote the function on Ω associating to each w ∈ Ω, the norm
in E of the element f (w) (resp. the norm in E ′ of the element ξ(w)).
Let f be a function on Ω with values in E such that f (Ω) is contained
in a finite dimensional subspace of E. We call such a function a finite
dimensional valued function. Let F be a finite dimensional subspace
of E containing f (Ω) and let e1 , e2 , . . . , en be a basis for F. Then there
exist n real valued functions α1 , α2 , . . . , αn on Ω such that ∀w ∈ Ω,
n
P
f (w) =
αi (w)ei . If S is any σ-algebra on Ω, it is clear that f ∈ S
i=1
if and only if ∀i = 1, 2, . . . n, αi ∈ S and that if f ∈ S and if S ⊂ Ôλ ,
then f is λ-integrable if and only if ∀i = 1, . . . n, αi is λ-integrable. If
f is λ-integrable, then a conditional expectation f C of f with respect to
1. Conditional expectations and disintegrations
14
C is defined as f C (w) =
13
n
P
i=1
αCi (w)ei ∀w ∈ Ω, where ∀i = 1, . . . n, αCi
is a conditional expectation of αi with respect to C . Note that αCi exist
∀i = 1, . . . , n. It can be easily seen that this definition is independent
of the choice of the basis of F and also in independent of the finite
dimensional subspace of E containing f (Ω).
Remark 5. If f is a finite dimensional valued λ-integrable function on
Ω with values in E and if ξ is any function on Ω with values in E ′ such
that ξ ∈ C (on E ′ we always consider the Borel σ-algebra of the strong
topology) and ξ is bounded in the sense that sup |ξ|(w) is a real number,
w∈Ω
then,
∀λ w, hξ, f iC (w) = hξ, f C i(w).
This is a consequence of the property (iv) of conditional expectations of extended real valued functions listed in § 5 of this chapter.
The alternative proof of the existence of conditional expectations of
Banach space valued functions depends on the following theorem.
Theorem 6. Let f be a finite dimensional valued λ-integrable function
on Ω with values in a Banach space E. Then,
∀λ w, | f C |(w) ≤ | f |C (w).
To prove this theorem, we need the following lemma.
Lemma 7. Let S be a σ-algebra on Ω.
(i) Let f be a step function on Ω with values in E, belonging to S.
Then there exists a function ξ : Ω → E ′ , ξ ∈ S, |ξ| ≤ 1 such that
∀w ∈ Ω, hξ(w), f (w)i = | f |(w).
(ii) If S ⊂ Ôλ and if f is any λ-integrable function on Ω with values in
E belonging to S, then there exists a sequence (ξn )n∈N of functions
on Ω with values in E ′ , ξn ∈ S and |ξn | ≤ 1 ∀ n ∈ N, such that
∀λ w, lim hξn (w), f (w)i = | f |(w).
n→∞
7. Another way of proving the existence...
15
14
Proof.
(i) Let f , a step function be of the form
n
P
i=1
χAi xi , Ai ∈ S
∀i = 1, . . . , n, Ai ∩ A j = ∅ if i , j, xi ∈ E ∀i = 1, . . . , n. By
Hahn-Banach theorem, there exists ∀i, i = 1, . . . , n, an element
n
P
χAi · xi .
ξi ∈ E ′ such that hξi , xi i = ||xi || and ||ξi || ≤ 1. Let ξ =
i=1
Then it is easily seen that ξ has all the required properties.
(ii) Let f be an arbitrary λ-integrable function belonging to S. Then
there exist a sequence ( fn )n∈N of step functions belonging to S
such that
∀λ w, | fn − f |(w) → 0 as n → ∞.
By (i) ∀n, ∃ξn : Ω → E ′ , ξn ∈ S, |ξn | ≤ 1 such that hξn (w),
fn (w)i = | fn |(w)∀w ∈ Ω.
hξn (w), f (w)i − | f |(w)
= hξn (w), f (w) − fn (w)i + hξn (w), fn (w)i − | f |(w)
= hξn (w), f (w) − fn (w)i + | fn |(w) − | f |(w)
≤ | f − fn |(w) + | fn − f |(w)
= 2| f − fn |(w).
Hence, ∀λ w, lim hξn (w), f (w)i exists and is equal to | f |(w).
n→∞
Proof of the theorem 6. Applying the above lemma (1, § 7, 7) to f C ,
we see that since f C ∈ C , there exists a sequence (ξn )n∈N of functions
on Ω with values in E ′ such that ξn ∈ C ∀n ∈ N, |ξn | ≤ 1 ∀n ∈ N and
∀λ w, lim hξn (w), f C (w)i = | f C |(w).
n→∞
By remark (1, § 7, 5),
∀ n ∈ N, ∀λ w, hξn (w), f C (w)i = hξn , f iC (w).
15
1. Conditional expectations and disintegrations
16
Therefore, ∀ n ∈ N, ∀λ w, |hξn (w), f C (w)i| = |hξn , f iC (w)| and
∀ n ∈ N, ∀λ w, |hξn , f iC |(w) ≤ |hξn , f i|C (w) by property (ii) of the conditional expectations of extended real valued functions, listed in § 5 of
this chapter.
Now, ∀ n ∈ N, |hξn , f i| ≤ ξn | · | f | ≤ | f | and hence, ∀ n ∈ N, ∀λ w,
C
C
|hξn , f i|C (w) ≤ | f |C (w).
Hence ∀λ w, ∀n ∈ N, |hξn , f i| (w) ≤ | f | (w).
C
C
C
Hence, ∀λ w, ∀n ∈ N, hξn (w), f (w)i ≤ | f | (w). Hence, ∀λ w, | f |(w) ≤
| f |C (w).
Remark 8. Actually, one can prove that if f is a finite dimensional valued λ-integrable function, belonging to a σ-algebra S contained in Ôλ ,
then there exists a function ξ on Ω with values in E ′ , ξ ∈ S, |ξ| ≤ 1 such
that ∀w ∈ Ω, hξ(w), f (w)i = | f |(w). But this is very difficult. Note that
once this is proved, the proof of theorem (1, § 7, 6) follows more easily.
Now, let us turn to the proof of the existence theorem of conditional
expectations for Banach space valued λ-integrable functions.
Let m be the vector subspace of all finite dimensional valued functions belonging to Ôλ and λ-integrable. Then m is dense in L1 (Ω; O;
λ; E), as we can approximate any f ∈ L1 (Ω; O; λ; E) by step functions.
Consider the linear map uO,C from m to L1 (Ω; C ; λ; E) given by
Z
16
uO,C ( f ) = f C .
Z
|uO,C ( f )|(w)dλ(w) =
| f C |(w)dλ(w).
By theorem (1, § 7, 6). ∀λ w, | f C |(w) ≤ | f |C (w).
Hence
Z
Z
| f C |(w)dλ(w) ≤
| f |C (w)dλ(w)
Z
=
| f |(w)dλ(w).
Hence, ||uO,C ( f )|| ≤ || f || where ||uO,C ( f )|| (resp. || f ||) denotes the
norm of uO,C ( f ) (resp. norm of f ) in L1 (Ω; C ; λ; E) (resp. in L1 (Ω; O;
λ; E)).
8. A few properties of conditional...
17
Hence uO,C is a contraction linear map and therefore, there exists a
unique extension of this map to the whole of L1 (Ω; O; λ; E) which we
again denote by uO,C . It can be easily seen that uO,C ( f ) is a conditional
expectation of f with respect to C .
8 A few properties of conditional expectations of
Banach space valued integrable functions
As before, let (Ω, O, λ) be a measure space, C a σ-algebra contained in
Ôλ and let λ restricted to C be σ-finite. Let E be a Banach space over
the real numbers.
Proposition 9. If f ∈ L1 (Ω; O; λ; E), then
∀λ w, | f C |(w) ≤ | f |C (w).
Proof. There exists a sequence ( fn )n∈N of step functions belonging to
Ôλ such that
(i) fn → f in L1 (Ω; O; λ; E)
(ii) ∀λ w, fn (w) → f (w) in E
and
(iii) ∀λ w, fnC (w) → f C (w) in E.
Since fn → f in L1 (Ω; O; λ; E), | fn | → | f | in L1 (Ω; O; λ). Hence
| fn → | f |C in L1 (Ω; C ; λ). Therefore, there exists a subsequence fnk
such that ∀λ w, | fnk |C (w) → | f |C (w) in E. Since
|C
∀λ w, | fnCk |(w) ≤ | fnk |C (w),
passing to the limit, we see that
17
∀λ w, | f C |(w) ≤ | f |C (w).
1. Conditional expectations and disintegrations
18
Proposition 10. Let fn be a sequence of functions on Ω, belonging to
Ôλ with values in E such that fn converges to a function f in the sense
of the dominated convergence theorem, i.e., ∀λ w, fn (w) → f (w) in E
and there exists a non-negative real valued λ-integrable function g on Ω
such that ∀λ w, ∀n ∈ N, | fn |(w) ≤ g(w). Then fnC converges to f C also
in the sense of the dominated convergence theorem.
Proof. Without loss of generality let us assume that f ≡ 0, and fn (w) →
0 for all w ∈ Ω.
Let c◦ (E) denote the vector space of all sequences x = (xn )n∈N , xn ∈
E ∀ n ∈ N and xn → 0 in E as n → ∞. Define ∀x ∈◦ (E), |||x||| =
sup ||xn || where ||xn || is the norm of the element xn in E. Then, it is easily
n
seen that x → |||x||| is a norm in c◦ (E) and this norm makes c◦ (E), a
Banach space.
Let h be a function on Ω with values in c◦ (E). Then there exists a
sequence (hn )n∈N of functions on Ω with values in E such that ∀w ∈ Ω,
h(w) = (h1 (w), h2 (w), . . . , hn (w), . . .).
It can be easily seen that if S is any σ-algebra on Ω, then h ∈ S if
and only if ∀n ∈ N, hn ∈ S and that if S ⊂ Ôλ , then h is λ-integrable if
and only if ∀n ∈ N, hn is λ-integrable. Moreover, it can be easily seen
that if h ∈ Ôλ and is λ-integrable, then ∀A ∈ Ôλ .
Z
Z
Z
Z
h2 dλ, . . . dλ, . . .).
hdλ = ( h1 dλ,
A
18
A
A
A
Hence if h ∈ Ôλ and is λ-integrable, then
∀λ w, hC (w) = (hC1 (w), hC2 (w), . . . , hCn (w), . . .)
Consider the sequence (χn )n∈N of functions on Ω with values in
given by



0 if m < n
(χn )m (w) = 

 fm (w) if m ≥ n
c◦ (E)
8. A few properties of conditional...
19
where (χn )m (w) stands for the mth coordinate of χn (w).
Then ∀w ∈ Ω, χn (w) → 0 in c◦ (E). ∀ n ∈ N, χn ∈ Ôλ and is
λ-integrable since ∀n ∈ N, ∀λ w, |χn |(w) = sup | fm |(w) ≤ g(w) and g is
m≥n
λ-integrable.
Z
|χn |dλ =
Z
sup | fm |(w)dλ(w) ↓ 0 as n → ∞.
m≥n
Hence
χn → 0 in L1 (Ω; O; λ; c◦ (E)).
Therefore,
χCn → 0 in L1 (Ω, C ; λ; c◦ (E)).
Let us prove that ∀λ w, χCn (w) → 0 in c◦ (E). Now, ∀λ w, |χCn |(w)
is a decreasing function of n and hence lim |χCn |(w) exists ∀λ m. Let
n→∞
dw = lim |χCn |(w) when the limit of |χCn |(w) exists.
n→∞
Z
|χCn |(w)dλ(w) ↓
Z
dw dλ(w), as n → ∞.
Since χCn → 0 in L1 (Ω; C ; λ; c◦ (E)), it follows that
and hence ∀λ w, dw = 0.
Hence,
∀λ w, χCn (w) → 0 in c◦ (E).
R
dw dλ(w) = 0
Hence,
∀λ w, sup | fmC |(w) → 0 as n → ∞.
m≥n
This means that ∀λ w, fnC (w) → 0 in E as n → ∞. Since | fn | ≤ g, 19
∀λ w, | fn |C (w) ≤ gC (w) and gC is λ-integrable, since g is. Hence fnC
converges to zero, in the sense of the dominated convergence theorem.
Remark 11. When E = R, the above proposition (1, § 8, 10) is proved
in Doob [1] in the pages 23-24. Though the theory of conditional expectations for Banach space valued functions is not used there as is done
here, the idea is essentially the same.
20
1. Conditional expectations and disintegrations
The properties (v), (vi) and (vii) stated in § 5 of this chapter for functions belonging to L1 (Ω; O; λ) are also true for any f ∈ L1 (Ω; O; λ; E),
as can be easily seen. Property (iv) also is true with the same assumptions on g and on g f as there.
Thus, in this chapter, we have proved the existence and uniqueness
of conditional expectations for Banach space valued λ-integrable functions on a measure space (Ω, O, λ). We shall see in § 1 of Chapter 3,
that the existence of disintegration of λ is linked to the existence of conditional expectations for measure valued functions on Ω, as defined in
chapter 2.
Chapter 2
Measure valued Functions
1 Basic definitions: Fubini’s theorem for extended
real valued integrable functions
In this chapter, we shall study the measure valued functions.
20
Let (Ω, O, λ) be a measure space. Let (Y, Y ) be a measurable space.
Let m+ (Y, Y ) be the set of all positive measures on Y .
Definition 12. A measure valued function ν on Ω with values in m+ (Y,
Y ) is an assignment to each w ∈ Ω, a positive measure νw on Y .
If (λCw )w∈Ω is a disintegration of λ with respect to a σ-algebra C ⊂
Ôλ , it can be considered as a measure valued function λC on Ω with
values in m+ (Ω, O) taking w ∈ Ω to λCw .
If ν is a measure valued function on Ω with values in m+ (Y, Y ) and
if f is any non-negative function on Y belonging to Y , then ν( f ) will
denote the function on Ω taking w to νw ( f ). If B is a set belonging to
Y , ν(B) will stand for ν(χB ).
Definition 13. A measure valued function ν on Ω with values in m+ (Y,
Y ) is said to be measurable with respect to a σ-algebra S on Ω or is
said to belong to S if ∀ B ∈ Y , the extended real valued function ν(B)
belongs to S.
If ν belongs to S, we write ν ∈ S.
21
2. Measure valued Functions
22
21
If λC is a disintegration of λ with respect to a σ-algebra C ⊂ Ôλ ,
note that λC ∈ C .
If ν ∈ S, we see immediately that ∀ non-negative function f on Y,
f ∈ Y , ν( f ) ∈ S.
Definition 14. If ν is a measure valued function on Ω with values in
m+ (Y, Y ), belonging to Ôλ , the integral of ν with respect
to λ is defined
R
as the measureRJ on Y given by ∀B ∈ Y , J(B) = νw (B)dλ(w). It is
written as J = νw dλ(w). If A ∈ O, the integral of ν over A is defined
as the integral of ν with
R respect to the measure χA · λ. The integral of ν
over A is written as νw dλ(w).
A
Note that if λC is a disintegration of λ with respect toR a σ-algebra
C ⊂ Ôλ , the integral of λC with respect to λ is λ. i.e. λ = λCw dλ(w).
Note also that from our definition, it easily follows that if J is the
integral of ν with
R respect to λ, then for every function f on Y, f ≥ 0,
f ∈ Y , J( f ) = νw ( f )dλ(w). Hence, ∀ f ≥ 0, f ∈ Y , J( f ) = 0 implies
that ∀λ w, νw ( f ) = 0. In particular, if B ∈ Y is such that its J-measure is
zero, then ∀λ w its νw -measure is also zero.
Note that it also follows easily from our definition that if f is an
extended real valued function on Y, f ∈ Y and J-integrable, then ∀λ w, it
is νw -integrable. Moreover, the function w → νw ( f ) (defined arbitrarily
on the set of points w where
f is not νw -integrable) belongs to Ôλ and is
R
λ-integrable. Further, R νw ( f )dλ(w) = J( f ).
R
If A ∈ O, note that νw dλ(w)( f ) is equal to νw ( f )dλ(w), for every
A
A
function f on Y, f ≥ 0 and f ∈ Y .
The following theorem and the corollary (2, § 1, 16) contains as a
special case, as we shall see towards the end of this chapter, the usual
Fubini’s theorem. Hence we shall call this also as Fubini’s theorem.
22
Theorem 15 (Fubini). Let S be a σ-algebra contained in Ôλ . Let ν be
a measure valued function on Ω with values in m+ (Y, Y ) belonging to
S and with integral J. Let f be an extended real valued function on Y,
f ≥ 0 belonging to YˆJ . Then,
(i) ∀λ w, f is νw -measurable, i.e., f ∈ Yˆνw .
1. Basic definitions: Fubini’s theorem...
23
(ii) The function w → νw ( f ) (defined arbitrarily on the set of w ∈ Ω
for which f is not νw -measurable) ∈ Ŝλ and
R
(iii)
νw ( f )dλ(w) = J( f ).
Proof.
(i) Since f ≥ 0 belongs to YˆJ , there exist functions fi on
Y, fi ≥ 0, fi ∈ Y , i = 1, 2, such that f1 ≤ f ≤ f2 everywhere
and the set B = {y ∈ Y | f1 (y) , f2 (y)} is of J-measure zero. Since
J(B) = 0, ∀λ w, νw (B) = 0. i.e., there exists a set A ∈ Ôλ such
that λ(A) = 0 and if w < A, νw (B) = 0. Since f1 ∈ Y , it follows
therefore that if w < A, f is νw -measurable i.e., f ∈ Yˆνw .
(ii) Since for i = 1, 2, w → νw ( fi ) ∈ S, it follows that w → νw ( f ) ∈
Ŝλ .
R
R
(iii) νw ( f )dλ(w) = νw ( fi )dλ(w)
= J( fi )
= J( f )
Corollary 16 (Fubini). With S, ν and J as in the above theorem, if f is
a J-integrable extended real valued function on Y, f ∈ YˆJ , then
(i) ∀λ w, f is νw -measurable, i.e., f ∈ Yˆνw , and is νw -integrable
(ii) the function w → νw ( f ) (defined arbitrarily on the set of w ∈ Ω for
which f is not νw -integrable), belongs to Ŝλ and is λ-integrable,
and
R
(iii)
νw ( f )dλ(w) = J( f ).
Proof. If f = f + − f − with the usual notation, then the corollary immediately follows from the above theorem by applying it to f + and f − . Corollary 17. Let C be a σ-algebra contained in Ôλ . Let (λCw )w∈Ω be a 23
disintegration of λ with respect to C . Let f be an extended real valued
function on Ω, f ≥ 0 (resp. f λ-integrable) belonging to Ôλ . Then,
(resp. f is λCw -measurable
(i) ∀λ w, f is λCw -measurable i.e. f ∈ ÔλC
w
C
and λw -integrable).
24
2. Measure valued Functions
(ii) the function w → λCw ( f ) {defined arbitrarily on the set of w ∈ Ω,
for which f is not λCw -measurable (resp. f is not λCw -integrable)}
belongs to Cˆλ and is λ-integrable and
R
(iii)
λCw ( f )dλ(w) = λ( f ).
Proof. When f is ≥ 0 (resp. f is λ-integrable), this follows immediately
from theorem (2, § 1, 15) (resp. from the above CorollaryR (2, § 1, 16))
by taking λC instead of ν and observing that λC ∈ C and λCw dλ(w) =
λ.
We saw is § 3 of Chapter 1, how the existence of a disintegration
of λ with respect to C implies immediately the existence of
conditional expectations for functions f ≥ 0 on Ω, f ∈ O and that
w → λCw ( f ) is a conditional expectation of f with respect to C . If f is
≥ 0 on Ω and belongs to Ôλ , we see from the above Corollary (2, § 1,
17) that ∀λ w, f is λCw -measurable and the function w → λCw ( f ) belongs
to Cˆλ . It can be easily checked that the function w → λCλ ( f ) is actually
a conditional expectation of f with respect to Cˆλ and hence is almost
everywhere equal to a conditional expectation of f with respect to C .
Similar result holds again when f ∈ Ôλ and is λ-integrable.
Thus, we see how the existence of a disintegration of λ with respect
to a σ-algebra C contained in Ôλ , implies as well the existence of conditional expectations with respect to C for λ-integrable extended real
valued functions belonging to Ôλ .
In § 2, we shall extend the results of this section to Banach space
valued λ-integrable functions.
(λCw )w∈Ω
24
2 Fubini’s theorem for Banach space valued integrable functions
Throughout this section, we fix a measure space (Ω, O, λ), a σ-algebra
S ⊂ Ôλ , a measurable space (Y, Y ), a measure valued function ν on
Ω with values in m+ (Ym, Y ) belonging to S, having an integral J with
respect to λ and a Banach space E over the real numbers.
2. Fubini’s theorem for Banach space...
25
Proposition 18. Let g be a step function on Y with values in E belonging
to YˆJ (resp. Y ). Let further, g be J-integrable.
Then
(i) ∀λ w, g ∈ Yˆνw and is νw -integrable (resp. ∀λ w, g is νw -integrable).
(ii) The function w → νw (g) with values in E (defined arbitrarily, on
the set of points w ∈ Ω where g is not νw -integrable, in such a
way that ∀w, νw (g) is still an element of E) belongs to Ŝλ (resp.
belongs to S) and is λ-integrable and
R
(iii)
νw (g)dλ(w) = J(g).
Proof.
(i) Let g =
n
P
χAi · xi where for i = 1, . . . n, Ai ∈ YˆJ (resp.
i=1
Aj =
Ai ∈ Y ), Ai ∩
∅ if i , j and xi ∈ E. By theorem (2, § 1,
15), ∀i, ∀λ w, Ai ∈ Yˆνw . Hence ∀λ w, ∀i, Ai ∈ Yˆνw . Therefore, ∀λ w,
g ∈ Yˆνw .
Since g is J-integrable, ∀i, J(Ai ) < +∞. Hence ∀λ w, ∀i, νw (Ai ) <
+∞. Hence ∀λ w, g is νw -integrable.
(ii) Let A = {w ∈ Ω | g is νw − integrable }. By (i) λ is carried by A
and ∀w ∈ A,
n
X
νw (g) =
νw (Ai )xi
i=1
∀n ∈ N, ∀w ∈ Ω, let gn (w) =
n
P
inf(νw (Ai ), n)xi .
i=1
Then, ∀n ∈ N gn is a finite dimensional valued function belonging 25
to Ŝλ (resp. belonging to S). Further, gn (w) → νw (g) in E ∀w ∈
A. Since λ is carried by A and since, ∀n ∈ N gn ∈ Ŝλ (resp.
gn ∈ S and A ∈ S) it follows that the function w → νw (g) belongs
to Ŝλ (resp. belongs to S).
∀i, w → νw (Ai ) is λ-integrable, since J(Ai ) < +∞, and hence
w → νw (g) is λ-integrable, since for w ∈ A,
|νw (g)| ≤
n
X
i=1
|νw (Ai )| · ||xi ||.
2. Measure valued Functions
26
(iii)
R
R P
n
νw (g)dλ(w) = ( νw (Ai )xi )dλ(w)
i=1
n R
P
=
νw (Ai )dλ(w) · xi
=
i=1
n
P
J(Ai )xi
i=1
= J(g)
Corollary 19. Let C be a σ-algebra contained in Ôλ and let (λCw )w∈Ω
be a disintegration of λ with respect to C . Let g be a step function on Ω
with values in E, g ∈ Ôλ (resp. g ∈ O) and λ-integrable. Then
and is λ-integrable (resp. ∀λ w, g is λCw -integrable).
(i) ∀λ w, g ∈ ÔλC
w
(ii) The function w → λCw (g) with values in E (defined arbitrarily, on
the set of points w ∈ Ω where g is not λCw -integrable, in such a
way that ∀w ∈ Ω, λCw (g) is still an element of E) belongs to Cˆλ
(resp. belongs to C ) and is λ-integrable and
R
(iii)
λCw (g)dλ(w) = λ(g).
Proof. This follows immediately from the above proposition (2, R§ 2,
18), by taking λC instead of ν and observing that λC ∈ C and λCw
dλ(w) = λ.
26
In the following theorem, let us consider the case of an arbitrary Jintegrable function on Y with values in E. Since this theorem contains
as a special case, the usual Fubini’s theorem, as we shall see below, we
call it also as Fubini’s theorem.
Theorem 20 (Fubini). Let f be a function on Y with values in E, f ∈ YˆJ
(resp. f ∈ Y ) and J-integrable. Then,
(i) ∀λ w, f ∈ Yˆνw and is νw -integrable (resp. ∀λ w, f is νw -integrable).
(ii) The function w → νw ( f ) on Ω with values in E (defined arbitrarily
on the set of points w ∈ Ω where f is not νw -integrable) belongs
to Ŝλ (resp. belongs to S) and is λ-integrable and
2. Fubini’s theorem for Banach space...
(iii)
R
27
νw ( f )dλ(w) = J( f ).
Proof. Since f ∈ YˆJ (resp. Y ) and is J-integrable, there exists a
sequence (gn )n∈N of step functions on Y with values in E and a nonnegative real valued function g on Y belonging to Y and J-integrable
such that ∀n, gn ∈ YˆJ (resp. ∀n, gn ∈ Y ) and J-integrable, ∀J y,
|gn |(y) ≤ g(y), ∀J y, gn (y) → f (y) in E and J(gn ) → J( f ) in E.
(i) By proposition (2, § 2, 18), ∀n, ∀λ w, gn ∈ Yˆνw . Since ∀J y,
gn (y) → f (y) in E, ∀λ w, ∀νw y, gn (y) → f (y) in E. Hence, ∀λ w,
f ∈ Yˆνw . Further, by the same proposition (2, § 2, 18), ∀n, ∀λ w,
gn is νw -integrable. Hence, ∀λ w, ∀n, gn is νw -integrable.
Also ∀λ w, g is νw -integrable.
Since ∀J y, ∀n, |gn |(y) ≤ g(y), we have ∀λ w, ∀νw y, ∀n, |gn |(y) ≤
g(y) and hence,
∀λ w, ∀n ∈ N, νw (|gn |) ≤ νw (g) < +∞.
Hence, by Fatou’s lemma, ∀λ w, f is νw -integrable, and by the
dominated convergence theorem,
27
∀λ w, νw(gn ) → νw ( f ) in E.
(ii) By proposition (2, § 2, 18), ∀n ∈ N, the function w → νw (gn )
belongs to Ŝλ (resp. belongs to S) and since ∀λ w, νw (gn ) → νw (g)
in E (resp. since the set of w where νw (gn ) converges to νw ( f )
belongs to S and carries λ) it follows that w → νw ( f ) also belongs
to Ŝλ (resp. belongs to S).
Since ∀λ w, νw (gn ) → νw ( f ) in E and since ∀λ w, ∀n ∈ N, |νw (gn )|
≤ νw (|gn |) ≤ νw (g) and since w → νw (g) is λ-integrable, it follows
by Fatou’s lemma again that w → νw ( f ) is λ-integrable and again
by the dominated convergence theorem,
Z
Z
νw (gn )dλ(w) →
νw ( f )dλ(w).
2. Measure valued Functions
28
J(gn ) → J( f ) in E.
R
But
R ∀n ∈ N, J(gn ) = νw (gnR)dλ(w) by proposition (2, § 2, 18).
Since νw (gn )dλ(w) converges to νw ( f )dλ(w), it follows that
(iii)
J( f ) =
Z
νw ( f )dλ(w).
Corollary 21. Let C be a σ-algebra contained in Ôλ , and let (λCw )w∈Ω
be a disintegration of λ with respect to C . Let f be a function on Ω with
values in E, f ∈ Ôλ (resp. f ∈ O) and λ-integrable. Then
and is λCw -integrable
(i) ∀λ w, f ∈ ÔλC
w
(ii) The function w → λCw ( f ) with values in E (defined arbitrarily on
the set of w ∈ Ω where f is not λCw -integrable in such a way that
λCw ( f ) still takes values in E∀w ∈ Ω) belongs to Cˆλ (resp. belongs
to C ) and is λ-integrable. and
R
(iii)
λCw ( f )dλ(w) = λ( f ).
28
Proof. This follows immediately from the above theorem (2, § 2,R 20)
by applying it to λC instead of ν and observing that λC ∈ C and λCw
dλ(w) = λ.
We shall now deduce the usual Fubini’s theorem from the above
theorem (2, § 2, 20).
Let (X, X, µ) and (Z, z, ν) be two measure spaces with µ (resp. ν) σfinite on X (resp. on z). Let ∀x ∈ X, δx ⊗ ν be the product measure of
δx and ν on the σ-algebra X ⊗ z on the set X × Z. Then x → δ x ⊗ ν is a
measure valued function on X with values in m+ (X × Z, X ⊗ z), belonging
to X and has the measure µ ⊗ ν for its integral with respect to µ.
Let f be a function on X × Z with values in a Banach space E,
d
ˆ µ⊗ν . Then, by the above theorem (2, § 2, 20).
f ∈X
⊗z
2. Fubini’s theorem for Banach space...
29
ˆ δx ⊗ν and is δx ⊗ ν integrable; i.e., ∀µ x, the function
(i) ∀λ x, f ∈ X⊗z
z → f (x, z) belongs to ẑν and is ν-integrable and
Z
∀ν x, | f |(x, z)dν(z) < +∞.
R
(ii) x → δx ⊗ ν( f ) belongs to X̂µ and is µ-integrable i.e., x → f (x, z)
dν(z) is µ-integrable and
R
R
(iii)
(δx ⊗ ν)( f )dµ(x) = f dµ ⊗ ν
Z Z
Z
i.e. ( f (x, z)dν(z))dµ(x) =
f (x, z)dµ ⊗ ν(x, z).
When E = R, this is the usual Fubini’s theorem.
It is now clear how to deduce the Fubini’s theorem for functions
which are non-negative (resp. integrable) and extended real valued from
theorem (2, § 1, 15) (resp. Corollary (2, § 1, 16)).
Form Corollary (2, § 2, 21), we see that if f is a J-integrable function
on Y with values in a Banach space E, f ∈ YˆJ (resp. f ∈ Y ) and if C is
a σ-algebra contained in Ôλ and if (λCw )w∈Ω is a disintegration of λ with 29
respect to C , then the almost everywhere defined function w → λCw ( f )
belongs to Cˆλ (resp. C ) and is λ-integrable. One can easily check that
w → λCw ( f ) is actually a conditional expectation of f with respect to
Cˆλ (resp. with respect to C ) first by considering step functions and
then extending to f . Thus w → λCw ( f ) is almost everywhere equal to
a conditional expectation of f with respect to C (resp. is a conditional
expectation of f with respect to C ).
Thus, we see how the existence of a disintegration implies the existence of conditional expectations for Banach space valued integrable
functions as well. Hence the importance of the existence of disintegrations. In the next chapter, we shall give some sufficient conditions for
the existence of disintegration of a measure with respect to a σ-algebra.
Chapter 3
Conditional expectations of
measure valued functions,
Existence and uniqueness
theorems
1 Basic definition
In this chapter, we shall define the notion of conditional expectation 30
for measure valued functions, and prove some existence and uniqueness
theorems. Our results, as we shall see, will give immediately an important consequence of a result of M. Jirina [1] on regular conditional
probabilities.
Let (Ω, O, λ) be a measure space. Let C be a σ-algebra contained
in Ôλ . Let λ restricted to C be σ-finite. Let (Y, Y ) be a measurable
space. Let ν be a measure valued function on Ω with values in m+ (Y, Y ),
ν ∈ Ôλ .
Definition 22. A measure valued function νC on Ω with values in m+ (Y,
Y ) is said to be a conditional expectation of ν with respect to C if
(i) A νC ∈ C and
31
3. Conditional expectations of measure valued...
32
(ii) ∀A ∈ C ,
R
νCw dλ(w) =
A
R
νw dλ(w).
A
Note that νC is a conditional expectation of ν if and only if for every
function f on Y, f ≥ 0, f ∈ Y , the function νC ( f ) ∈ C and
Z
Z
C
∀A ∈ C , νA ( f )dλ(w) =
νw ( f ) f λ(w).
A
31
A
Thus, νC is a conditional expectation of ν if and only if for every
function f on Y, f ≥ 0, f ∈ Y , the function νC ( f ) is a conditional
expectation of ν( f ).
We note also that if (λCw )w∈Ω is a disintegration of λ with respect to
C , then the measure valued function λC on Ω taking w ∈ Ω to λCw is a
conditional expectation with respect to C of the measure valued function
δ on Ω with values in m+ (Ω, O), taking w ∈ Ω to the measure δw (the
Dirac measure at w). And conversely, every conditional expectation of
the measure valued function δ is a disintegration of λ. Thus, we see how
the existence of disintegration for a measure is related to the existence
of conditional expectation of measure valued functions.
2 Preliminaries
Before we proceed to prove the existence and uniqueness theorems of
conditional expectations for measure valued functions, we collect below
in the subsections § 2.1, § 2.2, § 2.3 and § 2.4, some important theorems,
propositions and definitions which will be used in the proofs of the main
theorems of this chapter.
2.1 The monotone class theorem and its consequences
Let X be a non-void set. Let S be a class of subsets of X.
We say S is a π-system on X if it is closed with respect to finite
intersections.
We say S is a d-system on X if
(i) X ∈ S ,
2. Preliminaries
33
(ii) A, B ∈ S , A ⊂ B ⇒ B\A ∈ S and
(iii) ∀ n ∈ N, An ∈ S , An ↑⇒
S
An ∈ S .
n∈N
If C is any class of subsets of X, we shall denote by d(C ) (resp.
σ(C )) the smallest d-system (resp. σ-algebra) containing C . d(C )
(resp. σ(C )) will be called the d-system (resp. σ-algebra) generated
by C .
In these Notes, the following theorem, whenever it is referred to will
always be referred to as the Monotone class theorem.
Theorem (Monotone class thoerem). If S is a π-system, d(S ) =
σ(S ).
For a proof of this theorem, see R.M. Blumenthal and R.K. Getoor
[1]. Chap. 0, § 2, page 5, theorem (2.2).
Proposition 23. Let (X, K) be a measurable space. Let S be a π-system 32
containing X and generating X. Let µ and ν be two positive finite measures on X. If µ and ν agree on S , then µ and ν are equal.
Proof. The class C = {A ∈ X | µ(A) = ν(A)} is a d-system containing
S . Hence C ⊃ d(S ). But by the Monotone class theorem, d(S ) is
the σ-algebra generated by S which is X. Hence C ⊃ X and therefore
C = X and thus µ and ν are equal.
Proposition 24. Let µ and ν be two positive σ-finite measures on a measurable space (X, X). Let S ⊂ X be a π-system generating X and conS
Bn = X, µ(Bn ) = ν(Bn ) < +∞∀ n ∈
taining a sequence (Bn )n∈N with
n∈N
N. If µ and ν agree on S , then µ and ν are equal.
Proof. First, let us fix a n ∈ N.
Consider the class C = {A ∈ X | µ(A ∩ Bn ) = ν(A ∩ Bn )}. Then C
is a d-system containing S . Since S is a π-system and generates X, by
the Monotone class theorem C ⊂ X. Hence,
∀ A ∈ X, µ(A ∩ Bn ) = ν(A ∩ Bn ).
3. Conditional expectations of measure valued...
34
Since n is arbitrary, we have
∀ n ∈ N, ∀A ∈ X, µ(A ∩ Bn ) = ν(A ∩ Bn ).
if An =
n
S
Bi, note that
i=1
∀ n, ∀A ∈ X, µ(A ∩ An ) = ν(A ∩ An ).
Since
An ↑ X, we see that ∀A ∈ X, µ(A) = ν(A).
2.2 The lifting theorem of D. Maharam
33
Let (X, X, µ) be a measure space with µ σ-finite on X. Let L∞ (X; X; µ)
stand for the Banach space whose underlying vector space is the vector
space of all µ-equivalence classes of extended real valued functions on
X, belonging to X̂µ , having a finite essential supremum and the norm is
the essential supremum. Let B(X; X) stand for the Banach space whose
underlying vector space is the vector space of all real valued bounded
functions on X belonging to X̂µ and the norm is the supremum.
In the following, we have to keep in mind the fact that when we say
f is an element of L∞ (X; X; µ), we mean by f not a single function on
X, but a class of functions on X, any two functions of a class differing
only on a set of µ-measure zero at most. Thus if g is a function on X
and f ∈ L∞ (X; X; µ), the meaning of ‘g ∈ f ’ is clear i.e. g belongs to
the class f . If h is a bounded function on X, ∈ X̂µ , h ∈ will denote the
unique element of L∞ (X; X; µ) to which h belongs. Thus, if ‘a’ is a real
number, considered as the constant function ‘a’ on X, the meaning of a
is clear. If f ∈ L∞ (X; X; µ), we say f is non-negative and write f ≥ 0 if
there exists a function g on X, g ∈ f and a set Ng ∈ X̂µ with µ(Ng ) = 0
such that g ≥ 0 on ∁Ng . Note that if there exists one function on g ∈ f
having this property viz. there exists a set Ng with µ(Ng ) = 0 and g ≥ 0
on ∁Ng , then every function belonging to f also has this property. If
f1 and f2 are two elements ∈ L∞ (X; X; µ), we say f1 is greater than or
equal to f2 and write f1 ≥ f2 , if f1 − f2 ≥ 0.
2. Preliminaries
35
The following important theorem is due to D. Maharam and whenever we refer this theorem, we will be referring it as the lifting theorem.
For a proof of this theorem see D. Maharam [1].
34
Theorem D. Maharam. The Lifting Theorem. There exists a mapping
ρ from L∞ (X; X; µ) to B(X; X) such that
(i) ρ is linear and continuous
(ii) ρ( f ) ∈ f ∀ f ∈ L∞ (X; X; µ)
(iii) ρ( f ) ≥ 0 if f ≥ 0 and
(iv) ρ(1) ≡ 1
Such a mapping ρ is called a lifting.
2.3 Some theorems on L∞ (X; X; µ)
In this section, let (X; X, µ) be a measure space with µ(X) < +∞.
∞
to µ as
R If f ∈ L (X; X; µ), we define the integral of f with respect
R
gdµ where g is any function belonging to f . Note that gdµ exists
since µ is a finite measure and g is essentially bounded. Note also that
the integral of f with respect to µ is independent of the choice of the
function g Rchosen to belong to f . The integral of f with respect to µ is
written as f dµ.
Definition 25. Let ( fi )i∈I be a family of elements of L∞ (X; X; µ). An
element f ∈ L∞ (X; X; µ) is said to be a supremum of the family ( fi )i∈I
in the L∞ -sense if
(i) f ≥ fi ∀i ∈ I and
(ii) g ∈ L∞ (X; X; µ), g ≥ fi ∀i ∈ I ⇒ g ≥ f .
Note that through a supremum need not always exist, it is unique if
it exists.
If the supremum of a family ( fi )i∈I , fi ∈ L∞ (X; X; µ) ∀i, exists in the
L∞ -sense, we say SUPµ fi exists and denote the supremum by SUPµ fi .
i∈I
i∈I
3. Conditional expectations of measure valued...
36
If I is a finite set say {1, . . . n}, then we write SUPµ ( f1 , . . . , fn ) instead of
SUP µ fi . If f ∈ L∞ (X; X; µ), we denote by || f ||∞ , the essential suprei∈{1,...,n}
mum of f .
35
Proposition 26. If ( fn )n∈N is an increasing sequence of elements ∈
L∞ (X; X; µ) such that it is bounded in norm i.e. sup || fn ||∞ < +∞, then
n
R
R
SUPµ fn exists and (SUPµ fn )dµ = sup fn dµ.
n∈N
n∈N
Proof. Choose ∀ n ∈ N, a function hn on X, hn ∈ X̂µ such that hn ∈ fn .
Since ( fn )n∈N is increasing, ∀n ∈ N, ∃ a set En ∈ X̂µ such that µ(En ) = 0
and if x < En , hn (x) ≤ hn+1 (x).
∞
S
Let E =
En . Then E ∈ X̂µ and µ(E) = 0. If x < E, h1 (x) ≤ h2 (x)
n=1
. . . ≤ hn (x) ≤ hn+1 (x) ≤ . . .. Thus, if x < E, (hn (x))n∈N is a monotonic
non-decreasing sequence of extended real numbers and hence, ∀x < E,
lim hn (x) exists.
n→∞
Define



lim hn (x) if x < E

n→∞
h(x) = 


0
if x ∈ E.
Then h ∈ X̂µ h ∈ L∞ (X; X; µ) since sup || fn ||∞ < +∞.
n
∞
It is clear
R that h is theR supremum of ( fn )n∈N in the L -sense.
Since h dµ = lim hn dµ, it follows that
n→∞
Z
(SUPµ fn )dµ = sup
n∈N
n∈N
Z
fn dµ.
Proposition 27. Let ( fi )i∈I be a directed increasing family of elements
of L∞ (X; X; µ) such that the family is bounded in norm, i.e. sup || fi ||∞ <
i∈I
+∞. Then, SUPµ fi exists.
i∈I
Proof. Since µ is a finite measure and since
Z
sup || fi ||∞ < +∞, sup
fi dµ < +∞.
i∈I
i∈I
2. Preliminaries
Let α = sup
i∈I
Since ( fi )i∈I
37
R
fi dµ. Then α ∈ R.
is a directed increasing family, we can find an increasing
Rsequence ( fn )n∈N such that ∀n ∈ N, fn belongs to the family ( fi )i∈I , and
fn dµ ↑ α. From the previous proposition, SUPµ fn exists. Let f =
n∈N
SUPµ fn .
n∈N
Let us show that f is the supremum of ( fi )i∈I in the L∞ -sense.
Fix an i ∈ I.
It is easily seen that
Z
Z
SUPµ ( fi , fn )dµ ↑
SUPµ ( fi , f )dµ as n → ∞.
Z
Z
∀ n ∈ N, SUPµ ( fi , fn )dµ ≤ sup
f j dµ,
j∈I
since ( fi )i∈I is a directed increasing family.
Hence,
Z
∀ n ∈ N,
Hence,
But
Z
R
SUPµ ( fi , fn )dµ ≤ α.
SUPµ ( fi , f )dµ ≤ α.
SUPµ ( fi , f )dµ ≥ α, since SUPµ ( fi f ) ≥ f and
Hence,
Z
Z
SUPµ ( fi f )dµ = α =
R
f dµ = α.
f dµ.
This shows that f ≥ fi .
Since i ∈ I is arbitrary, it follows that f ≥ fi ∀i ∈ I. Now, let
g ∈ L∞ (X; X; µ) be such that g ≥ fi ∀i ∈ I. Then g ≥ fn ∀n ∈ N and
hence g ≥ f .
This shows that f is the supremum of ( fi )i∈I in the L∞ -sense.
Proposition 28. Let ( fi )i∈I be a directed increasing family of elements of
L∞ (X; X; µ) and let f ∈ L∞ (X; X; µ). Then, the following are equivalent,
(i) f = SUPµ fi
i∈I
36
3. Conditional expectations of measure valued...
38
(ii) f ≥ fi ∀i ∈ I and there exists a sequence ( fn )n∈N , ∀n ∈ N fn
belonging to the family ( fi )i∈I such that f = SUPµ fn .
n∈N
(iii) f ≥ fi ∀i ∈ I and
R
f dµ = sup
i∈I
R
fi dµ.
Proof. (i) =⇒ (ii). Since f = SUPµ fi and f ∈ L∞ (X; X; µ), it follows
i∈I
R
that the family ( fi )i∈I is bounded in norm. Let α = sup fi dµ.
i∈I
37
Then α ∈ R. Then, if ( fn )n∈N is an increasing sequence of elRements, ∀ n ∈ N, fn belonging to the family ( fi )i∈I such that
fn dµ ↑ α, we can show as in the proof of the previous proposition (3, § 2.3, 27) that f = SUPµ fn .
n∈N
(ii) =⇒ (iii). LetR hn = SUPµ ( f1 , f2 , . . . , fn ). Then ∀ n ∈ N, hn ∈ L∞ (X;
X; µ) and hn dµ ↑ f dµ.
R
R
Since
∀
i
∈
I,
f
≥
f
,
f
dµ
≥
f
dµ
∀i
∈
I,
and
hence,
f dµ ≥
i
i
R
sup fi dµ.
i∈I
On the other hand, ∀ n ∈ N,
R
hn dµ ≤ sup
i∈I
family ( fi )i∈I is directed increasing).
Hence
Z
f dµ = sup
i∈I
Z
R
fi dµ, (since the
fi dµ.
(iii) =⇒ (i). Since f ∈ L∞ (X; X; µ) and f ≥ fi ∀i ∈ I, the family ( fi )i∈I
is bounded in norm.
R
PR
fi dµ. Then α ∈ R. Let (gn )n∈N be an
Let α = f dµ =
i∈I
increasing sequence of elements belonging to LR∞ (X; X; µ) such
that ∀ n ∈ N, gn belongs to the family ( fi )i∈I and gn dµ ↓ α.
Let g = SUPµ gn (SUPµ gn exists because of proposition (3, § 2.3,
n∈N R
n∈N
R
26)). Then g dµ = α = f dµ. R Since f ≥
R fi ∀i ∈ I, f ≥
gn ∀ n ∈ N, and hence f ≥ g. Since f dµ = g dµ, it follows
that f = g.
2. Preliminaries
39
If h ∈ L∞ (X; X; µ) is such that h ≥ fi ∀i ∈ I, then h ≥ gn ∀n ∈ N
and since g = f = SUPµ gn , it follows that h ≥ f . Hence f =
SUPµ fi .
Corollary 29. Let ( fi )i∈I be a directed increasing family of elements of
L∞ (X; X; µ) and let f ∈ L∞ (X; X; µ). Then f = SUPµ fi if and only if
i∈I
(i) f ≥ fi ∀i ∈ I and
(ii) there exists an increasing sequence
R ( fn )n∈NR, fn , ∀ n ∈ N, belonging to the family ( fi )i∈I such that fn dµ ↑ f dµ.
Proof. This is an immediate consequence of the above proposition.
Proposition 30. Let ρ be a lifting from L∞ (X; X; µ) to B(X; X). Let 38
( fi )i∈I be a directed increasing family of elements of L∞ (X; X; µ) bounded in norm. Let f = SUPµ fi . Then,
i∈I
ρ( f ) ∈ SUPµ ρ( fi ) and ∀µ x, ρ( f )(x) = sup ρ( fi )(x).
i∈I
i∈I
In particular,
sup ρ( fi ) ∈ X̂µ .
i∈I
Proof. Note that ρ( f ) ∈ SUPµ ρ( fi ) is clear since SUPµ ρ( fi ) = SUPµ fi
i∈I
i∈I
(since ρ( fi ) ∈ fi ∀i ∈ I), ρ( f ) ∈ f and f = SUPµ fi . Since ∀ i ∈ I, f ≥ fi ,
i∈I
ρ( f )(x) ≥ ρ( fi )(x) for all x ∈ X and hence
ρ( f )(x) ≥ sup ρ( fi )(x)
i∈I
for all x ∈ X.
Let ( fn )n∈N be an increasing sequence of elements, ∀ n ∈ N, fn
belonging to the family ( fi )i∈I such that f = SUPµ fn . (Such a sequence
n∈N
exists because of the proposition (3, § 2.3, 28)).
3. Conditional expectations of measure valued...
40
R
R
R
R
Then ρ( fn ) ↑ and fn dµ ↑ f dµ. Hence, ρ( fn )dµ ↑ ρ( f )dµ.
Since ρ( fu ) ↑, by the monotone convergence theorem,
Z
Z
sup ρ( fn )dµ.
sup ρ( fn )dµ =
n
n
Hence,
R
ρ( f )dµ =
R
sup ρ( fn )dµ. But sup ρ( fn )(x) ≤ ρ( f )(x) for
n
n
all x ∈ X, since ∀ n ∈ N, ρ( fn )(x) ≤ ρ( f )(x) for all x ∈ X. Hence ∀µ x,
ρ( f )(x) = sup ρ( fn )(x).
n
Now, for all x ∈ X.
ρ( f )(x) ≥ sup ρ( fi )(x) ≥ sup ρ( fn )(x).
i∈I
n
Hence, ∀λ x, ρ( f )(x) = sup ρ( fi )(x). Since ρ( f ) ∈ X̂µ and since ∀µ x,
i∈I
sup ρ( fi )(x) is equal to ρ( f )(x), it follows that
i∈I
sup ρ( fi ) ∈ X̂µ .
i∈I
39
Let z be a sub σ-algebra of X̂µ . Let f ∈ L∞ (X; X; µ). Let g be any
function on X, g ∈ f . Since ∀µ x, g(x) ≤ || f ||∞ and µ is a finite measure, it
follows that g is µ-integrable, and hence gz , a conditional expectation of
g with respect to z exists. ∀µ x, |gz (x)| ≤ || f ||∞ and hence gz is essentially
bounded. Since any two conditional expectations of g with respect to z
are equal µ-almost everywhere, we have a unique element of L∞ (X; z; µ)
to which any conditional expectation of g with respect to z belongs. Note
that this element of L∞ (X; z; µ) is independent of the function g, chosen
to belong to f and hence depends only on f . We denote this element by
f z and call it the conditional expectation of f with respect to z.
Proposition 31. Let z be a sub σ-algebra of X̂µ . Let ( fi )i∈I be a directed
increasing family of elements of L∞ (X; X; µ) bounded in norm. Let f =
SUPµ fi . Then SUPµ fiz exists and is equal to f z .
i∈I
i∈I
2. Preliminaries
41
Proof. Since ( fi )i∈I is a directed increasing family bounded in norm,
( fiz )i∈I is also directed increasing and bounded in norm. Hence SUPµ fiz
i∈I
exists.
To prove that f z = SUPµ fiz , it is sufficient to prove, because of
i∈I
R
R
proposition (3, § 2.3, 28), that f z ≥ fiz ∀i ∈ I and f z dµ = sup fiz dµ.
i∈I R
R
But
this
follows
immediately,
since
f
≥
f
∀i
∈
I,
f
dµ
= f z dµ,
i
R
R
R z
R
fi dµ = fi dµ∀i ∈ I, and f dµ = sup fi dµ.
i∈I
2.4 Radon Measures
Let X be a topological space. (By topological spaces in these Notes, we
always mean only non-void, Hausdorff topological spaces). Let X be its
Borel σ-algebra i.e. the σ-algebra generated by all the open sets of X.
Definition 32. A positive measure µ on X is is said to be a Radon measure on X if
(i) µ is locally finite i.e. every point x ∈ X has a neighbourhood V x 40
such that µ(V x ) < +∞. and
(ii) µ is inner regular in the sense that
∀ B ∈ X, µ(B) = sup µ(K)
K⊂B
K compact
Definition 33. Let µ be a Radon measure on a topological space X. Let
(Ki )i∈I be a family of compact sets of X and N, a µ-null set. {(Ki )i∈I , N}
is said to be a µ-concassage of X if
(i) X = N ∪
S
Ki
i∈I
(ii) Ki ∩ K j = ∅ if i , j, and
3. Conditional expectations of measure valued...
42
(iii) the family (Ki )i∈I is locally countable in the sense that every point
has a neighbourhood which has a non-void intersection with at
most a countable number of Ki ’s.
The following important theorem is stated here without proof; and
for a proof see L. Schwartz [2], page 48, theorem 13.
Theorem. If µ is a Radon measure on topological space X, there exists
a µ-concassage {(Ki )i∈I , N} of X.
Proposition 34. Let µ be a finite Radon measure on X. Then there exists
a µ-concassage {(Ki )i∈I , N} of X with I countable.
Proof. Since µ is a finite Radon measure, using the inner regularity of
µ, we see that there exists a µ-null set N1 and a sequence (Xn )n∈N of
compact sets of X such that
[
X = N1 ∪
Xn .
n∈N
41
Let {(K j ) j∈J , N2 } be a µ-concassage of X. Since the family (K j ) j∈J
is locally countable, every compact set can have a non-void intersection
only with at most a countable number of (K j ) j∈J . Hence, ∀ n ∈ N, there
exists a countable set In ⊂ J such that if j < In , Xn ∩ K j = ∅. Let
S
In . Then it is clear that {(Ki )i∈I , N} is a µ-concassage of X where
I=
n∈N
N = N1 ∪ N2 . Note the I is countable.
3 Uniqueness Theorem
Let (X, X) be a measurable space. Let µ be a positive measure on X.
Definition 35. X is said to have the µ-countability property if there exists a set N ∈ X with µ(N) = 0 such that the σ-algebra X ∩ ∁N on
X ′ = X ∩ ∁N consisting of sets of the form A ∩ ∁N, A ∈ X is countably
generated.
Examples . Suppose X is countably generated as in the case when X is
the Borel σ-algebra of a topological space X having the 2nd axiom of
3. Uniqueness Theorem
43
countability, then obviously X has the µ-countability property for any
positive measure µ on X. Also, it can be easily proved that if X̂µ is
countably generated, then X has the µ-countability property.
From now onwards, in this section, let (Ω, O, λ) be a measure space
and let (Y, Y ) be a measurable space. Let ν be a measure valued function
on Ω with values in m+ (Y, Y ), ν ∈ Ôλ . Let C beR a σ-algebra on Ω,
C ⊂ Ôλ . Let λ restricted to C be σ-finite. Let J = νw dλ(w).
Theorem 36 (The Uniqueness theorem). Let J be a σ-finite measure
on Y and let Y have the J-countability property. Then, if νC1 and νC2
are any two conditional expectations of ν with respect to C , we have
∀λ w.(νC1 )w = (νC2 )w where (νCi )w for i = 1, 2 stands for the measure
associated by νCi to w.
R
R
Proof. First note that J = νC1 dλ = νC2 dλ. Since J is σ-finite, it
therefore follows that ∀λ w, (νC1 )w and (νC2 )w are both σ-finite i.e. ∃ a
set N1 ∈ Ôλ with λ(N1 ) = 0 such that if w < N1 , (νC1 )w and (νC2 )w are
σ-finite measures on Y .
42
Since Y has the J-countability property there exists a set N ∈ Y
with J(N) = 0 such that the σ-algebra Y ′ = Y ∩ ∁N on Y ′ = Y ∩ ∁N
is countably generated. Since J(N) = 0, it follows that ∀λ w, (νC1 )w (N) =
(νC2 )w (N) = 0. i.e., there exists a set N2 ∈ Ôλ with λ(N2 ) = 0 such that
if w < N2 , (νC1 )w (N) = (νC2 )w (N) = 0.
With our assumptions that J is σ-finite and Y ′ is countably generated, we can find a class B of subsets of Y ′ such that B is countable,
B generates Y ′ and J(B) < +∞ ∀B ∈ B. Let C be the class of
subsets of Y ′ formed by the sets which are finite intersections of sets
belonging to B. Then C is countable, C is a π-system generating Y ′
and J(C) < +∞ ∀C ∈ C . There exists a set N3 ∈ Ôλ with λ(N3 ) = 0
such that if w < N3 , (νC1 )w (C) and (νC2 )w (C) are both finite for all C ∈ C .
Now ∀B ∈ Y , ∀λ w, (νC1 )w (B) = (νC2 )w (B) since both νC1 (B) and
C
ν2 (B) are conditional expectations with respect to C of the extended
real valued function ν(B). Hence in particular, ∀C ∈ C , ∀λ w, (νC1 )w
(C) = (νC2 )w (C).
Since C is countable, we therefore have,
∀λ w, ∀C ∈ C , (νC1 )w (C) = (νC2 )w (C) i.e.
3. Conditional expectations of measure valued...
44
there exists a set N4 ∈ Ôλ with λ(N4 ) = 0 such that if w < N4 ,
(νC1 )w (C) = (νC2 )w (C)
43
for all C ∈ C .
Let w < N1 ∪ N3 ∪ N4 . Then both (νC1 )w and (νC2 )w are σ-finite,
C
(ν1 )w (C) = (νC2 )w (C) ∀ C ∈ C and (νC1 )w (C) < +∞, (νC2 )w (C) < +∞
for all C ∈ C . Since C is a π-system generating Y ′ and countable,
by proposition (3, § 2.1, 24), we conclude that (νC1 )w = (νC2 )w on Y ′ .
Hence if w < N1 ∪ N3 ∪ N4 , the measures (νC1 )w and (νC2 )w are equal on
Y ′.
Therefore, if w < N1 ∪ N3 ∪ N4 ∪ N2 , the measures (νC1 )w and (νC2 )w
are equal on Y since if w < N2 , (νC1 )w (N) = (νC2 )w (N) = 0. Hence
∀λ w, (νC1 )w = (νC2 )2 on Y .
4 Existence theorems
Let X be a topological space and let X be its Borel σ-algebra. Let µ be
a positive measure on X.
Definition 37. X is said to have the µ-compacity (resp. µ-compacity
metrizability) property if there exists a set N ∈ X with µ(N) = 0 and a
sequence (Kn )n∈N of compact sets (resp. compact metrizable sets) such
S
Kn ∪ N and ∀ n ∈ N, µ(Kn ) < +∞.
that X =
n∈N
Note that if µ is a σ-finite Radon measure on X, X has the µ compacity property. If further to µ being a σ-finite Radon measure,
either X is metrizable or every compact subset of X is metrizable, then
X has the µ-compacity metrizability property. In particular, if X is a
Suslin space and µ is a σ-finite Radon measure on X, then X has the
µ-compacity metrizability property. (In a Suslin space, every compact
subset is metrizable).
We shall now give some sufficient conditions for the existence of
conditional expectations of measure valued functions.
4. Existence theorems
45
Throughout this section, we shall adopt the following notations.
(Ω, O, λ) is a measure space. Y is a topological space and Y is its
Borel-σ-algebra. ν is a measure valued function on Ω with values in
m+ (Y, Y ), ν ∈ Ôλ and having an integral J with respect to λ.
Theorem 38. Let a sequence (Kn )n∈N of compact sets of Y and a set
N ∈ Y exist with the following properties:
(i) Y =
S
Kn ∪ N
n∈N
(ii) J(N) = 0 and
(iii) ∀ n ∈ N, the restriction of J to Kn is a Radon measure on Kn .
Let C be a σ-algebra contained in Ôλ such that C is complete with
respect to λ and let λ restricted to C be σ-finite. Then, a conditional
expectation of ν with respect to C exists.
Proof. Let us split the proof in two cases case 1 and case 2. In case
1, we assume that Y is compact and J is a Radon measure on Y. In
this case, the assumptions (i), (ii) and (iii) mentioned in the statement of
the theorem are trivially verified. In case 2, we shall consider a general
topological space Y, having the properties (i), (ii) and (iii) mentioned in
the statement of the theorem. We shall deduce case 2 from case 1.
The proof in case 1 proceeds in three steps, Step 1, Step 2 and Step
3.
In Step 1, we define a measure valued function νC on Ω with values
in m+ (Y, Y ) such that ∀w ∈ Ω, νCw is a Radon measure on Y and such that
∀ real valued continuous function C on Y, w → νCw (ϕ) is a conditional
expectation with respect to C of the function w → νw (ϕ).
In Step 2, we prove that ∀ U open in Y, νC (χU ) is a conditional
expectation with respect to C of the function ν(χU ).
In Step 3, we prove that ∀B ∈ Y , νC (χB ) is a conditional expectation
with respect to C of the function ν(χB ).
Case 1. Y a compact space and J, a Radon measure on Y.
44
3. Conditional expectations of measure valued...
46
Step 1. Since J is a Radon measure on Y, J(Y) < +∞. Hence ∀λ w,
νw (Y) < +∞. Without loss of generality, we can assume that ∀ w ∈ Ω,
νw (Y) < +∞. For, consider the measure valued function ν′ on Ω with
values in m+ (Y, Y ) given by



νw if νw (Y) < +∞
ν′w = 

0 otherwise i.e. the zero measure if νw (Y) = +∞.
45
Then ν′ is a measure valued function on Ω with values in m+ (Y, Y ),
∈ Ôλ , ν′ has the same integral J as ν and further ∀ w ∈ Ω, ν′w is a
finite measure. Also, ∀λ w, ν′w = νw . Hence, if a conditional expectation
of ν′ with respect to C exists, it is a conditional expectation of ν with
respect to C , as well.
ν′
Hence, we may assume that ∀ w ∈ Ω, νw (Y) < +∞.
Let ϕ be any real valued bounded function on Y, ϕ ∈ Y . Since ∀w ∈
Ω, νw is a finite measure on Y , ∀ w ∈ Ω, ϕ is νw -integrable. Consider
the real valued function ν(ϕ) taking w to νw (ϕ). Since λ restricted to C
is σ-finite, a conditional expectation for ν(ϕ) with respect to C exists.
Let [ν(ϕ)]C be a conditional expectation of ν(ϕ) with respect to C .
Now, let us fix [ν(1)]C once and for all as the conditional expectation
of ν(1) with respect to C in such a way that ∀ w ∈ Ω, 0 ≤ [ν(1)]C (w) <
+∞.
Let ||ϕ|| be sup |ϕ(y)|
y∈Y
Then, |ν(ϕ)| ≤ ||ϕ||.ν(1).
Hence, ∀λ w, [ν(ϕ)]C (w) ≤ |ν(ϕ)|C (w) ≤ ||ϕ||[ν(1)]C (w). Hence, if
n
o
A = w : [ν(1)]C (w) = 0 , then A ∈ C and ∀λ w, w ∈ A, [ν(ϕ)]C (w) = 0.
Define the quotient
[ν(ϕ)]C
to be zero on the set A. On ∁A, the
[ν(1)]C
[ν(ϕ)]C
has a meaning.
[ν(1)]C
[ν(ϕ)]C
[ν(ϕ)]C
≤
||ϕ||.
Hence
the
function
(w)
is esNow, ∀λ w, [ν(1)]C
[ν(1)]C
quotient
4. Existence theorems
47
sentially bounded. Since it belongs to C ,
[ν(ϕ)]C
∈ L∞ (Ω; C ; λ).
[ν(1)]C
[ν(ϕ)]C
of L∞ (Ω; C ; λ) is independent of the
[ν(1)]C
choice of the conditional expectation [ν(ϕ)]C of ν(ϕ) and hence depends
only on ϕ.
Let ρ be a lifting from L∞ (Ω; C ; λ) to B(Ω; C ). The existence of 46
a lifting is guaranteed by the lifting theorem mentioned in § 2.2 of this
chapter.
Let C (Y) be the space of all real valued continuous functions on Y.
If w ∈ Ω, define the maps νCw on C (Y) taking real
 value as follows.
 [ν(ψ)]C 
 (w).
If ψ ∈ C (Y), define νCw (ψ) as [ν(1)]C (w). ρ 
[ν(1)]C 
Then, by the properties of the lifting νCw is a positive linear functional
on C (Y) and hence defines a positive Radon measure on Y. Let νC be
the measure valued function on Ω with values in m+ (Y, Y ) taking w to
νCw .


 [ν(ψ)]C 
Since C is complete, ∀ ψ ∈ C (Y), ρ 
 ∈ C and hence ∀
[ν(1)]C 
ψ ∈ C (Y), the function νC (ψ) belongs to C . Moreover, if B ∈ C ,
Note that this element
Z
νCw (ψ)dλ(w) =
Z
[ν(1)]C (w) · ρ
=
Z
[ν(1)]C ·
B
!
[ν(ψ)]C
(w)dλ(w)
[ν(1)]C
B
=
=
B
Z
B
Z
B
[ν(ψ)]C
dλ
[ν(1)]C
[ν(ψ)]C dλ
νw (ψ)dλ(w).
3. Conditional expectations of measure valued...
48
Hence νC (ψ) is a conditional expectation with respect to C of the
function ν(ψ).
Step 2. Let U be an open subset of Y, U , ∅. Since U is open, χU is a
lower-semi-continuous function on Y and hence there exists a directed
increasing family (ϕi )i∈I of continuous functions on Y such that ∀ i ∈ I,
0 ≤ ϕi ≤ χU and sup ϕi = χU .
i∈I
[ν(ϕi )]C
∀ i ∈ I, ∀λ w, (w) ≤ ||ϕi || ≤ 1.
[ν(1)]C
[ν(ϕi )]C
of elements of L∞ (Ω;
[ν(1)]C
[ν(ϕi )]C
exists by propoC ; λ) is bounded in norm and hence ( SUP)λ
[ν(1)]C
i∈I
sition (3, § 2.3, 27).
We claim that
Hence, the directed increasing family
47
SUPλ
i∈I
[ν(ϕi )]C
[ν(χU )]C
=
[ν(1)]C
[ν(1)]C
To prove this, it is sufficient to prove because of Corollary (3, § 2.3,
29) that
[ν(χU )]C
[ν(ϕi )]C
≥
∀i ∈ I,
[ν(1)]C
[ν(1)]C
and that there exists an increasing sequence (ϕn )n∈N of continuous functions, ϕn ∀n ∈ N, belonging to the family (ϕi )i∈I such that
Z
Z
[ν(χU )]C
[ν(ϕn )]C
dλ
=
dλ
lim
n→∞
[ν(1)]C
[ν(1)]C
Since ∀ i ∈ I, ϕi ≤ χU , we have
∀ i ∈ I, ν(ϕi ) ≤ ν′ (χU ).
Hence ∀ i ∈ I, ∀λ w, [ν(ϕi )]C (w) ≤ [ν(χU )]C (w). Therefore,
∀ i ∈ I,
[ν(χU )]C
[ν(ϕ1 )]C
≤
.
[ν(1)]C
[ν(1)]C
4. Existence theorems
49
Since J is a Radon measure, and (ϕi )i∈I is a directed increasing family of continuous functions with χU = sup ϕi , we have
i∈I
J(χU ) = sup J(ϕi ).
i∈I
Hence there exists an increasing sequence (ϕn )n∈N of continuous
functions, ∀ n ∈ N, ϕn belonging to the family (ϕi )i∈I such that
∀J y, ϕn (y) ↑ χU (y).
Hence,
48
∀λ w, ∀νw y, ϕn (y) ↑ χU (y),
Therefore,
∀λ w, νw (ϕn ) ↑ νw (χU ).
Since w → [γ(1)]C (w) ∈ C and A ∈ C , by property (iv) of the
conditional expectations of extended real valued functions mentioned in
§ 5 of Chapter 1 (actually the property of vO,C mentioned there is used),
we have
Z
Z
[νϕn ]C
[ν(ϕn )]C
(w)dλ(w)
=
(w)dλ(w)
∀ n ∈ N,
[ν(1)]C
[ν(1)]C
Z
νw (ϕn )
= ∁A
dλ(w)
[ν(1)]C (w)
∁A
Hence ∀ n ∈ N,
νw (ϕn ) ↑ νw (χU ),
Z
∁A
R
R [ν(ϕn )]C
νw (ϕn )
dλ =
dλ(w). Since ∀λ w,
[ν(1)]C
[ν(1)]C (w)
∁A
νw (ϕn )
dλ(w) ↑
[ν(1)]C (w)
Z
νw (χU )
dλ(w).
[ν(1)]C (w)
∁A
Again by the same property for vO,C mentioned in (iv) of the conditional expectations of extended real valued functions mentioned in § 5
of Chapter 1,
Z
Z
νw (χU )
[ν(χU )]C
dλ(w)
=
(w)dλ(w)
[ν(1)]C (w)
[ν(1)]C
∁A
∁A
3. Conditional expectations of measure valued...
50
Z
[ν(χU )]C
(w)dλ(w)
[ν(1)]C
Z
[ν(χU )]C
dλ
=
[ν(1)]C
=
R [ν(χU )]C
R [ν(ϕn )]C
dλ
exists
and
is
equal
to
dλ. This
n→∞
[ν(1)]C
[ν(1)]C
[ν(ϕi )]C
[ν(χU )]C
and therefore, by proposition (3,
=
SUP
proves that
λ
i∈I
[ν(1)]C
[ν(1)]C
§ 2.3, 30),




 [ν(χU )]C 
 [ν(ϕi )]C 
∀λ w, ρ 
 (w) = sup ρ 
 (w).
[ν(1)]C 
[ν(1)]C 
i∈I


 [ν(ϕi )]C 
and sup ρ 
 ∈ C since C is complete. Since ∀w ∈ Ω, νCw is a
C 
[ν(1)]
i∈I
Radon measure,
Hence, lim
49
νCw (χU ) = sup νCw (ϕi )
i∈I


 [ν(ϕi )]C 
 (w)
= sup[ν(1)] (w)ρ 
[ν(1)]C 
i∈I


 [ν(ϕi )]C 
C
= [ν(1)] (w). sup ρ 
 (w).
[ν(1)]C 
i∈I


 [ν(ϕi )]C 
C
C
 belong
Hence ν (χU ) ∈ C , since both [ν(1)] and sup ρ 
[ν(1)]C 
i∈I
to C .
Let B ∈ C . Then,


Z
Z
 [ν(ϕi )C ] 
C
C
νw (χU )dλ(w) = [ν(1)] (w) sup ρ 
 (w)dλ(w)
[ν(1)]C 
i∈I
C
B
B
=
ZB
B
[ν(1)]C (w)
[ν(χU )]C
(w)dλ(w)
[ν(1)]C
4. Existence theorems
51
=
Z
[ν(χU )]C (w)dλ(w)
=
ZB
νw (χU )dλ(w).
B
B
Hence νC (χU ) is a conditional expectation of ν(χU ).
Step 3. Let us prove that ∀ B ∈ Y , the function νC (χB ) belongs to C
and is a conditional expectation with respect to C of the function ν(χB ).
Consider the class C of all sets B ∈ Y for which νC (χB ) ∈ C and
is a conditional expectation with respect to C of the function ν(B). It is
easily seen that C is a d-system.
From Step 2, C contains the class U of all open sets of Y, U is 50
a π-system generating Y . Hence, by the Monotone class theorem, C
contains Y and hence C = Y .
Hence ∀B ∈ Y , the function νC (χB ) belongs to C and is a conditional expectation of the function ν(χB ) with respect to C . This shows
that the measure valued function νC on Ω with values in m+ (Y, Y ) is a
conditional expectation of ν with respect to C .
Thus, Case 1 is completely proved.
Case 2. Let Y be an arbitrary topological space having the properties
stated in the theorem, i.e. ∃ a sequence (Kn )n∈N of compact sets of Y
S
Kn ∪ N and such that
and a set N ∈ Y with J(N) = 0 such that Y =
n∈N
the measure JKn , the restriction of J to Kn is a Radon measure on Kn .
Let
Xn = Kn \(K1 ∪ . . . ∪ Kn−1 ).
Then ∀ n ∈ N, Xn is a Borel set of Y, Xn ∩ Xm = ∅ if n , m, Y =
S
Xn ∪N and the measure JXn , the restriction of J to Xn is a finite Radon
n∈N
measure on Xn . By proposition (3, § 2.4, 34), ∀ n ∈ N, ∃ a sequence
(Xnm )m∈N of mutually disjoint compact sets of Xn and a Borel set Nn of
Xn with JXn (Nn ) = 0 such that (Xnm )m∈N , Nn is a JXn -concassage of Xn .
3. Conditional expectations of measure valued...
52
Thus, there exists a sequence (Yn )n∈N of compact sets of Y and a set
S
Yn ∪ M, Yn ∩ Ym = ∅ if n , m
M ∈ Y with J(M) = 0 such that Y =
n∈N
and the measure JYn , the restriction of J to Yn is a Radon measure on Yn .
Let ∀n, Yn be the Borel σ-algebra of Yn .
Let ∀ n ∈ N, νn be the measure valued function on Ω with values in
m+ (Yn Yn ), taking w ∈ Ω to the measure νnw which is the restriction of νw
to Yn . Then, ∀ n ∈ N, νn ∈ Ôλ . ∀ B ∈ Yn ,
Z
Z
n
νw (B)dλ(w) =
νw (B)dλ(w) = J(B) = JYn (B).
51
Hence the integral of νn with respect to λ is the measure JYn which
is a Radon measure on Yn . Hence, by Case 1, ∀ n ∈ N, a conditional
expectation νC ,n of νn with respect to C exists.
Define ∀ n ∈ N, ∀ w ∈ Ω, the measures ν′ Cw ,n on Y as follows:
If B ∈ Y , define ν′ wC ,n (B) as equal to νwC ,n (B ∩ Yn ).
Then ∀ n ∈ N, ∀w ∈ Ω, ν′ wC ,n is a positive measure on Y and the me
assure valued function ν′ C ,n on Ω with values in m+ (Y, Y ) taking w to
ν′ wC ,n belongs to C .
P ′ C ,n
ν w i.e.
Define ∀ w ∈ Ω, the measure ν′ Cw on Y as ν′ Cw =
n∈N
P ′ C ,n
ν w (B). Then, the measure valued function ν′ C
∀B ∈ Y , ν′ Cw (B) =
n∈N
on Ω with values in m+ (Y, Y ) taking w to the measure ν′ Cw belongs to
C.
Let C ∈ C and B ∈ Y . Then,
Z X
Z
′C
ν w (B)dλ(w) = ( ν′ wC ,n (B))dλ(w)
C
C
=
n∈N
XZ
ν′ wC ,n (B)dλ(w)
n∈N C
=
XZ
νCw ,n (B ∩ Yn )dλ(w)
n∈N C
=
XZ
n∈N C
νnw (B ∩ Yn )dλ(w)
4. Existence theorems
53
since ∀ n ∈ N, νC ,n (B ∩ Yn ) is a conditional expectation with respect to
C of νn (B ∩ Yn ). So the left side above is
XZ
=
νw (B ∩ Yn )dλ(w)
n∈N
C
Z X
= ( νw (B ∩ Yn ))dλ(w).
C
n∈N
Since J(M) = 0, ∀λ w, νw (M) = 0. Hence, ∀B ∈ Y , ∀λ w,
P
νw (B ∩ 52
n∈N
Yn ) = νw (B) since the sequence (Yn )n∈N is mutually disjoint and Y =
S
Yn ∪ M.
n∈N
Hence,
Z X
C
n∈N
νw (B ∩ Yn )dλ(w) =
Z
νw (B)dλ(w).
C
Therefore, ν′ C (B) is a conditional expectation with respect to C of
the function ν(B)∀B ∈ Y . Hence ν′ C is a conditional expectation of ν
with respect to C .
Remark 39. The assumptions in the above theorem regarding Y and J
are fulfilled if J is a σ-finite Radon measure on Y and these are stronger
than the J-compacity property for Y.
Lemma 40. Let Y be a compact metrizable space. Let C be a σ-algebra
contained in Ôλ and let Cˆλ be the completion of C with respect to λ.
ˆ
Let νCλ be a measure valued function on Ω with values in m+ (Y, Y )
ˆ
belonging to Cˆλ such that ∀ w ∈ Ω, νCwλ is a Radon measure on Y and
ˆ
J ′ , the integral of νCλ with respect to λ be a finite measure on Y . Then,
there exists a measure valued function νC on Ω with values in m+ (Y, Y )
belonging to C , such that ∀w ∈ Ω, νCw is a Radon measure on Y and
ˆ
∀λ w, νCw = νCwλ .
Proof. Since Y is a compact metrizable space, the Banach space C (Y)
of all real valued continuous functions on Y has a countable dense set
3. Conditional expectations of measure valued...
54
D. If ψ ∈ C (Y), let ||ψ|| be sup |ψ(y)|. We can assume that D has the
y∈Y
53
following property, namely, given any f ∈ C (Y), f ≥ 0 and any ǫ > 0,
there exists a function ϕǫ ∈ D, ϕǫ ≥ 0 such that || f − ϕǫ || ≤ ǫ. i.e.,
the positive elements of C (Y) can be approximated at will by positive
elements of D. Let a sequence (ϕn )n∈N of continuous functions on Y,
constitute the set D.
ˆ
Now, ∀ n ∈ N, the function νCλ (ϕn ) belongs to Cˆλ and is λ-integrable. Hence ∀ n ∈ N, there exist functions f1n and f2n on Ω belonging to
C such that
ˆ
∀ w ∈ Ω, f1n (w) ≤ νCwλ (ϕn ) ≤ f2n (w)
n
o
and the set Bn = w ∈ Ω | f1n (w) , f2n (w) has λ-measure zero.
S
Bn . Then B ∈ C and λ(B) = 0.
Let B =
n∈N
Let ϕ ∈ C (Y). There exists a sequence ϕnk of functions belonging
to D such that
||ϕnk − ϕ|| → 0 as nk → ∞.
ˆ
ˆ
Since ∀ w, νCwλ is a Radon measure on Y and hence ∀w, νCwλ (Y) < +∞
ˆ
ˆ
since Y is compact, we have ∀w ∈ Ω, νCwλ (ϕnk ) → νCwλ (ϕ) as nk → ∞.
Hence, if w < B, lim f1nk (w) and lim f2nk (w) exist and both are
nk →∞
equal to lim
nk →∞
ˆ
νCwλ (ϕnk )
which is
nk →∞
ˆ
νCwλ (ϕ).
Therefore, ∀ϕ ∈ C (Y), ∀w < B, the lim f1nk (w) is independent of
nk →∞
the choice of the sequence (ϕnk ) chosen to converge to ϕ in C (Y).
Hence, ∀w ∈ Ω, define the map νCw on C (Y) as,



f1nk (w), if w < B

nlim
C
→∞
k
νw (ϕ) = 


0
if w ∈ B.
where ϕǫC (Y) and (ϕnk )nk ∈N is a sequence such that ∀nk ∈ N, ϕnk ∈ D
and ϕnk → ϕ as nk → ∞ in C (Y). ∀ w ∈ Ω, the map νCw defined on C (Y)
ˆ
as above is clearly linear and if w < B, νCw (ϕ) = νCwλ (ϕ) for all ϕ ∈ C (Y).
ˆ
ϕ
i.e. ∀λ w, ∀ϕ ∈ C (Y), γw (ϕ) = νCwλ (ϕ). Moreover the linear functional νCw
is positive because of our assumptions on D that the positive elements of
4. Existence theorems
55
C (Y), can be approximated by positive elements of D. Hence, ∀w ∈ Ω,
νCw is a Radon measure on Y.
ˆ
ˆ
Since ∀λ w, ∀ϕ ∈ C (ϕ) = νCwλ (ϕ) and since ∀w ∈ Ω, νCw and νCwλ
ˆ
are Radon measures, it follows that ∀λ w, the measures νCw and νCwλ are
equal, i.e.
54
Cˆλ
C
∀λ w, νw = νw .
We have to prove that the measure valued function νC on Ω taking
w to νCw belongs to C .
If ϕ ∈ C (Y), it is clear that the function νC (ϕ) belongs to C since ∀
n ∈ N, f1n ∈ C and B ∈ C .
Let U be a non-void open set. Since Y is metrizable, there exists an
increasing sequence ϕn of continuous functions on Y such that ∀n ∈ N,
0 ≤ ϕn ≤ χU and ϕn (y) ↑ χU (y) for all y ∈ Y.
Hence ∀w ∈ Ω, νCw (χU ) = lim νCw (ϕn ). Therefore, ∀U open, νC
n→∞
(χU ) ∈ C .
Now, a standard application of the Monotone class theorem will
yield that ∀C ∈ Y , the function νC (χC ) belongs to C .
Hence the measure valued function νC belongs to C and since ∀λ w,
ˆ
νCw = νCwλ , our lemma is completely proved.
Theorem 41. Let Y have the J-compacity metrizability property. Let
C be a σ-algebra contained in Ôλ and let λ restricted to C be σfinite. Then a conditional expectation of ν with respect to C exists and
is unique.
Proof. Let us prove the theorem under the assumption that Y is a compact metrizable space and J is a finite measure on Y . The general case
will follow along lines similar to case 2 of theorem (3, § 4, 38).
Since J is a finite measure on Y and Y is a compact metrizable
space, J is a Radon measure on Y. Let Cˆλ be the completion of C with
respect to λ. By case 1 of theorem (3, § 4, 38), a conditional expectation
ˆ
ˆ
νCλ of ν with respect to Cˆλ exists in such a way that ∀ w ∈ Ω, νCwλ is a
Radon measure on Y.
Z
Z
ˆ
νCwλ (Y)dλ(w) =
νw (Y)dλ(w) = J(Y) < +∞.
3. Conditional expectations of measure valued...
56
55
Hence, by the above lemma (3, § 4, 40), there exists a measure valued
function νC on Ω with values in m+ (Y, Y ) such that νC ǫC and ∀λ w,
ˆ
νCw = νCwλ .
ˆ
Hence, since νCλ is a conditional expectation of ν with respect to
Cˆλ .νC is a conditional expectation of ν with respect to C .
Since the Borel σ-algebra of a compact metrizable space is countably generated, we can easily see that if Y has the J-compacity metrizability property, Y has the J-countability property. Hence by theorem (3, § 3, 36) the conditional expectation of ν with respect to C is
unique.
5 Existence theorem for disintegration of a
measure. The theorem of M. Jirina
Throughout this section, let Ω be a topological space, O its Borel σalgebra and λ a positive measure on O.
We have already remarked in § 1 of this chapter, that a disintegration
of λ with respect to C is a conditional expectation of the measure valued
function δ on Ω with values in m+ (Ω, O) taking w to the Dirac measure
δw and vice versa. So, by theorem (3, § 3, 36) we get the following
uniqueness theorem for disintegrations and by the theorems (3, § 4, 38)
and (3, § 4, 41), we get the following two theorems for the existence of
disintegrations. More precisely, we have
Theorem 42 (Uniqueness). Let λ be a σ-finite measure on O and let
O have the λ-countability property. Let C be a σ-algebra
o contained in
n
C
and
Ôλ and let λ restricted to C be σ-finite. Then, if (λ1 )w
winOmega
o
n
are two disintegrations of λ with respect to C , we have ∀λ w,
(λC2 )w
w∈Ω
C
C
(λ1 )w = (λ2 )w .
Theorem 43 (Existence(M.Jirina)). Let there exist a sequence (Kn )n∈N
of compact sets of O and a set N ∈ O such that
56
(i) Ω =
S
n∈N
Kn ∪ N,
6. Another kind of existence theorem...
57
(ii) λ(N) = 0, and
(iii) ∀ n ∈ N, the restriction of λ to Kn is a Radon measure on Kn .
Let C be a complete σ-algebra contained in Ôλ i.e. let C = Cˆλ .
Then, a disintegration of λ with respect to C exists.
Theorem 44 (Existence). Let Ω have the λ-compacity metrizability
property. Let C be a σ-algebra contained in Ôλ and let λ|C be σ-finite,
where λ|C stands stands for the restriction on λ to C . Then, a disintegration of λ with respect to C exists and is unique.
We remark that the assumptions in the theorem (3, § 5, 43) are fulfilled if λ is a σ-finite Radon measure on Ω. Ω has the λ-compacity
metrizability property if λ is a σ-finite Radon measure on Ω and if Ω
is either metrizable or if every compact subset of Ω is metrizable. In
particular, if Ω is a Suslin space and if λ is a σ-finite Radon measure on
Ω, then Ω has the λ-compacity metrizability property.
When λ is a Radon probability measure on Ω i.e., λ is a Radon
measure and λ(Ω) = 1, the theorem (3, § 5, 43) is essentially due to
M. Jirina [1] in the sens that this theorem is an easy consequence of his
theorem 3.2, on page 448 and the ‘note added in proof’ in page 450.
6 Another kind of existence theorem for conditional
expectation of measure valued functions
Throughout this section, let Ω be a topological space, O its Borel σalgebra, λ a positive measure on O, Y a topological space and Y its
Borel σ-algebra.
If C is a σ-algebra of Ôλ , we saw in Chapter 1 how the existence of a
disintegration of λ with respect to C , implies immediately the existence
of conditional expectation with respect to C of non-negative, extended 57
real valued functions on Ω belonging to O and in Chapter 2, we saw
how the existence of a disintegration of λ with respect to C implies
the existence of conditional expectation with respect to Cˆλ of extended
real valued λ-integrable functions belonging to Ôλ and also of Banach
58
3. Conditional expectations of measure valued...
space valued λ-integrable functions belonging to Ôλ . In this section, we
shall consider the case of measure valued functions in relation to their
conditional expectations with respect to C , when a disintegration of λ
with respect to C exists.
Theorem 45. Let C be a σ-algebra contained in Ôλ . Let (λCw )w∈Ω be a
disintegration of λ with respect to C . Let ν be a measure valued function
on Ω with values in m+ (Y, Y ). Let ν ∈ O. Then the measure
valued
R
function νC on Ω with values in m+ (Y, Y ) defined as νCw = νw′ λCw (dw′ )
is a conditional expectation of ν with respect to C . In particular, a
conditional expectation of ν with respect to C exists.
Proof. Let f be a function on Y, f ≥ 0, f ∈ Y . Consider the extended
real valued function ν( f ). This function belongs to O since ν ∈ O. Since
C
R(λw )w∈Ω is a disintegration of λ with respect to C , the function w →
νw′ ( f )λCw (dw′ ) is a conditional expectation of the function ν( f ) with
C
respect to C . Hence the measure
R valued function ν on Ω with values
in m+ (Y, Y ) defined as νCw = νw , λCw (dw′ ) ∀w ∈ Ω, is a conditional
expectation of ν with respect to C .
58
When ν ∈ Ôλ , we cannot apply the above argument since for f ≥ 0
on Y, belonging to Y , ν( f ), though
R belongs to Ôλ , does not in general
belong to O. Hence, the integral νw′ ( f )λCw (dw′ ) does not have a meaning in general for all functions f on Y, f ≥ 0, f ∈ Y , since the measures
λCw are measures on O and not on Ôλ for all w ∈ Ω.
However, for functions ν belonging to Ôλ , we have the following
theorem of existence of conditional expectations.
Theorem 46. Let be a σ-algebra contained in Ôλ . Let (λCw )w∈Ω be a
disintegration of λ with respect to C . Let ν be a measure valued
R function
on Ω with values in m+ (Y, Y ), ν belonging to Ôλ . Let J = νw dλ(w).
Let J be σ-finite and let Y have the J-countability property. Then, a
conditional expectation of ν with respect to C exists and is unique.
For the proof of this theorem, we need the following lemma.
Lemma 47. Let ν be a measure valued
R function on Ω with values in
m+ (Y, Y ), ν belonging to Ôλ . Let J = νw dλ(w). Let J be σ-finite. Let
6. Another kind of existence theorem...
59
Y have the J-countability property. Then there exists a measure valued
function ν′ on Ω with values in m+ (Y, Y ) such that ν′ ∈ O and ∀λ w,
ν′w = νw on Y .
Proof. Since J is σ-finite, there exists an increasing sequence (En )n∈N
S
En and J(En ) < +∞∀ n ∈ N.
of sets belonging to Y such that Y =
n∈N
Hence,
∀ n ∈ N,
∀λ w, νw (En ) < +∞,
Hence,
∀λ w, ∀n ∈ N, νw (En ) < +∞.
i.e. ∃ a set N1 ∈ O with λ(N1 ) = 0 such that if w < N1 , νw (En ) < +∞
for all n ∈ N.
Let N ∈ Y with J(N) = 0 such that the σ-algebra Y ∩ ∁N on
Y ′ = Y ∩ ∁N is countably generated.
Since J(N) = 0, ∃ N2 ∈ O with λ(N2 ) = 0 such that if w < N2 ,
νw (N) = 0.
Let C = (Cn )n∈N , ∀n ∈ N, Cn ∈ Y ′ generate Y ′ . We can assume
that C is a π-system and that Y ′ ∈ C .
Consider ∀ n ∈ N, ∀ m ∈ N, the function w → νw (Em ∩ Cn ).
This function belongs to Ôλ and Em ∩ Cn ∈ Y . Therefore, there exist functions f1m,n and f2m,n on Ω belonging to O such that f1m,n (w) ≤ 59
νw (Em ∩ Cn ) ≤ f2m,n (w) for all w ∈ Ω and the set
n
o
N m,n = w ∈ Ω | f1m,n (w) , f2m,n (w)
has λ-measure zero.
S m,n
Let N3 =
N . Then N3 ∈ O and λ(N3 ) = 0.
m∈N
n∈N
If y0 is any point of Y, define the measure valued function ν′ on Ω
with values in m+ (Y, Y ) as follows.



νw , if w < N1 ∪ N2 ∪ N3
′
νw = 

δy0 , if w ∈ N1 ∪ N2 ∪ N3
From the definition, it is clear that ∀λ w, ν′w = νw . To prove the
lemma, we have to prove only that ν′ ∈ O. Now,
3. Conditional expectations of measure valued...
60
∀m ∈ N, ∀n ∈ N , ∀w ∈ Ω,
ν′w (Em ∩ Cn ) = χ∁{N1 ∪N2 ∪N3 } (w) · νw (Em ∩ Cn ) + χN1 ∪N2 ∪N3 (w) · χEm ∩Cn (y0 )
= χ∁{N1 ∪N2 ∪N3 } · f1m,n (w) + χN1 ∪N2 ∪N3 (w) · χEm ∩Cn (y0 ).
Since f1m,n ∈ O and N1 ∪ N2 ∪ N3 ∈ O, it is clear that the function
m ∩ C n ) belongs to O.
Now fix a m ∈ N. Let
ν′ (E
B = B ∈ Y ′ | ν′ (B ∩ Em ) belongs to O .
It is clear that B is a d-system. B contains the π-system C which
generates Y ′ . Hence, by the Monotone class theorem, B = Y ′ . Hence,
since m is arbitrary, ∀ m ∈ N, ∀ B ∈ Y ′ , ν′ (B ∩ Em ) belongs to O. Since
∀B ∈ Y ′ , ∀ w ∈ Ω, ν′w (B) = lim ν′w (B ∩ Em ), it follows that ν′ (B)
m→∞
belongs to O.
Let
A ∈ Y , A = A ∩ Y ′ ∪ A ∩ N.
60
If w < N1 ∪ N2 ∪ N3 , ν′w (A) = ν′w (A ∩ Y ′ ), since ν′w (A ∩ N) =
νw (A ∩ N) = 0. Hence,
ν′w (A) = χ∁{N1 ∪N2 ∪N3 } (w).ν′w (A ∩ Y ′ ) + χN1 ∪N2 ∪N3 (w).χA (y0 )
for all w ∈ Ω. Since A ∩ Y ′ ∈ Y ′ , the function ν′ (A ∩ Y ′ ) belongs to O.
Since N1 ∪ N2 ∪ N3 ∈ O, it follows from the above expression of ν′ (A),
that ν′ (A) belongs to O.
Since A ia an arbitrary set ∈ Y , it follows that the measure valued
function ν′ belongs to O.
Proof of theorem 46 From the above lemma (3, § 6, 47) there exists
a measure valued function ν′ on Ω with values in m+ (Y, Y ) such that
ν′ ∈ O and ∀λ w, ν′w = νw . From theorem (3, § 6, 45) there exists a
conditional expectation ν′ C with respect to C for ν′ , since ν′ ∈ O. Since
∀λ w, ν′w = νw , it is clear that ν′ C is also a conditional expectation of ν
with respect to C .
The uniqueness follows from theorem (3, § 3, 36).
7. Conditions for a given family...
61
Thus, we see, if a disintegration of λ with respect to C exists, how
theorem (3, § 6, 45) guarantees immediately the existence of conditional
expectations with respect to C of measure valued functions belonging to
O and how theorem (3, § 6, 46) guarantees the same for measure valued
functions belonging to Ôλ if J is σ-finite and if Y has the J-countability
property where J is the integral of ν with respect to λ. But theorems (3,
§ 5, 43) and (3, § 5, 44) give sufficient conditions for the existence of
disintegration of λ. Hence, we have the following existence theorem of
conditional expectations for measure valued functions.
Theorem 48. (i) Let C be a complete σ-algebra contained in Ôλ and
le there exist a sequence (Kn )n∈N of compact sets of Ω and a set N ∈ O
S
Kn ∪ N and the restriction of λ to
with λ(N) = 0 such that Ω =
n∈N
Kn ∀ n ∈ N is a Radon measure on Kn .
Or (ii) Let C be an arbitrary σ-algebra contained in Ôλ and let Ω 61
have the λ-compacity metrizability property.
Then, under either of the conditions (i) and (ii), if ν is any measure valued function on Ω with values in m+ (Y, Y ), ν ∈ O, a conditional
expectation of ν with respect
to C exists.
R
If ν ∈ Ôλ , if J = νw dλ(w), if J is σ-finite, and if Y has the Jcountability property, then under either of the conditions (i) and (ii), a
conditional expectation of ν with respect to C exists and is unique.
7 Conditions for a given family (λCw )w∈Ω of positive
measures on O, to be a disintegration of λ with
respect to C and consequences
Throughout this section, let (Ω, O, λ) be a measure space and let C be
a σ-algebra contained in Ôλ . Let (λCw )w∈Ω be a given family of positive
measures on O. We shall discuss below some necessary and sufficient
conditions for this family to be a disintegration of λ, with respect to C .
Proposition 49. For (λC )w∈Ω to be a disintegration of λ, it is necessary
and sufficient that the following three conditions are verified.
3. Conditional expectations of measure valued...
62
(i) The measure valued function λC taking w ∈ Ω to λCw , belongs to
C
R
(ii) λ = λCw dλ(w)
and
(iii) ∀ A ∈ C , ∀λ w, λCw is carried by A or by ∁A according as w ∈ A
or w ∈ ∁A.
Proof. Necessity. Let (λCw )w∈Ω be a disintegration of λ with respect to
C . Then, by the definition of disintegration, (i) and (ii) are fulfilled. Let
us verify (iii).
Let A ∈ C . λC (∁A) is a conditional expectation of χ∁A with respect
to C . Therefore,
Z
Z
χ∁A (w)dλ(w) =
λCw (∁A)dλ(w)
A
i.e. 0 = λ(A ∩ ∁A) =
A
Z
λCA (∁A)dλ(w).
A
62
Therefore, ∀λ w, χA (w). λCA (∁A) = 0. Similarly,
∀λ w, χ∁A (w).λCw (A) = 0.
This proves that ∀λ w, λCw (∁A) = 0 if w ∈ A and λCw (A) = 0, if
w ∈ ∁A. Hence, ∀λ w, λCw is carried by A if w ∈ A and is carried by ∁A
if w ∈ ∁A.
Sufficiency. We have to prove that ∀B ∈ O. λC (B) is a conditional
expectation of χB with respect to C . i.e., we have to prove that ∀ A ∈ C ,
Z
Z
χB dλ =
λC (B)dλ;
A
A
i.e. we have to prove that
λ(A ∩ B) =
Z
A
λC (B)dλ.
7. Conditions for a given family...
Z
λCw (B)dλ(w)
=
A
Z
λCw (B
63
Z
∩ A)dλ(w) +
A
λCw (B ∩ ∁A)dλ(w).
∁A
Since ∀λ w, w ∈ A, λCw is carried by A,
∀λ w, w ∈ A, λCw (B ∩ ∁A) = 0.
Hence,
Z
λCw (B ∩ ∁A)dλ(w) = 0.
A
Therefore,
Z
λCw (B)dλ(w)
=
Z
λCw (A ∩ B)dλ(w).
A
A
Since
λ=
λ(A ∩ B) =
=
Z
Z
Z
λCw dλ(w),
λCw (A ∩ B)dλ(w)
λCw (A
∩ B)dλ(w) +
A
Z
λCw (A ∩ B)dλ(w).
∁A
Since ∀λ w, λCw is carried by ∁A if w ∈ ∁A,
Z
λCw (A ∩ B)dλ(w) = 0.
63
∁A
Hence
λ(A ∩ B) =
=
Z
A
Z
λCw (A ∩ B)dλ(w)
λCw (B)dλ(w)
A
64
3. Conditional expectations of measure valued...
Definition 50. Let (X, X ) be a measurable space. Let x ∈ X. The Xatom of x is defined to be the intersection of all sets belonging to X and
containing x.
Definition 51. Let (X, X) be a measurable space. We say X is countably
separating if there exists a sequence (An )n∈N , An ∈ X ∀n ∈ N, such that
∀x ∈ X, the X-atom of x is the intersection of all the An ’s that contain x.
It can be easily proved that if X is countably generated, then it is
countably separating.
Proposition 52. Let (i) the measure valued function λC on Ω taking w
to λCw belong to C .
R
(i) λ = λCw dλ(w) and
(ii) ∀λ w, λCw is carried by the C -atom of w.
Then (λCw )w∈Ω is a disintegration of λ.
Proof. Let (λCw )w∈Ω have (i), (ii) and (iii). From condition (iii), we see
by the definition of the C -atom of w, given A ∈ C , ∀λ w, λCw is carried
by A if w ∈ A and λCw is carried by ∁A if w ∈ ∁A. Thus, the condition
(iii) of this proposition implies the condition (iii) of the proposition (3,
§ 7, 49). Hence, the conditions (i), (ii) and (iii) of this proposition imply
the conditions (i), (ii) and (iii) of the proposition (3, § 7, 49). Hence
(λCw )w∈Ω is a disintegration of λ with respect to C
64
Proposition 53. Let C be countably separating. Then, the conditions
(i), (ii) and (iii) of the previous proposition (3, § 7, 52) are necessary for
(λCw )w∈Ω to be a disintegration of λ with respect to C .
Proof. Let C be countably separating and let (λCw )w∈Ω be a disintegration of λ with respect to C . Then (i) and (ii) are obvious. We have to
only verify that ∀λ w, λCw is carried by the C -atom of w.
Since C is countably separating, by definition, there exists a sequence (An )n∈N of sets belonging to C such that ∀w ∈ Ω, the C -atom of
w is the intersection of all the An ’s that contain w.
7. Conditions for a given family...
65
By condition (iii) of the proposition (3, § 7, 49), ∀ n ∈ N, ∃ a set
Nn ∈ O with λ(Nn ) = 0 such that if w < Nn , λCw is carried by An if
w ∈ An and is carried by ∁An if w < An .
S
Nn . Then N ∈ O and λ(N) = 0. Let w < N and let
Let N =
n∈N
Aw be the C -atom of w. Since C is countably separating, there exists a
sequence (nk ) of natural numbers such that
[
Aw =
Ank ,
Ank ∋w
Ank belonging to the sequence (An )n∈N ∀nk .
For every nk , since w ∈ Ank , λCw is carried by Ank and hence is carried
T
Ank .
by Aw since Aw =
Ank ∋w
Since w < N is arbitrary, it follows that ∀λ w, λCw is carried by the
C -atom of w.
The following counter example will show that the condition (iii) of
the proposition (3, § 7, 52), namely ∀λ w, λCw is carried by the C -atom
of w is not necessarily true for a disintegration (λCw )w∈Ω of λ with respect
to C , without further assumptions on C .
Let Ω be the circle S 1 in R2 . Let O be the Borel σ-algebra of Ω. 65
Let λ be the Lebesgue measure of S 1 on O. Let C be the σ-algebra of
all symmetric Borel sets. Since Ω is a compact metric space and λ is
a Radon probability measure, by theorem (3, § 5, 44), a disintegration
(λCw )w∈Ω of λ with respect to C exists and is unique. Consider the family
(δCw )w∈Ω of measures given by δCw = 12 (δw + δ−w ). It is easy to check that
(δCw )w∈Ω is a disintegration of λ with respect to C . Since the disintegration is unique, we have ∀λ w, λCw = δCw , i.e. ∀λ w, ∀Cw = 12 (δw + δ−w ).
Let Cˆ be the σ-algebra generated by C and all the λ-null sets of O.
ˆ
i.e. let Cˆ = C Vηλ . By theorem (3, § 5, 44), a disintegration (λCw )w∈Ω of
λ with respect to Cˆ exists and is unique. It is clear that (λCw )w∈Ω is also
a disintegration of λ with respect to Cˆ. Hence, by uniqueness,
ˆ
∀λ w, λCw = λCw =
1
(δw + δ−w ).
2
3. Conditional expectations of measure valued...
66
If w ∈ Ω, the Cˆ-atom of w is the single point w itself. Since we see
ˆ
ˆ
that ∀λ w, λCw = 21 (δw + δ−w ), ∀λ w, λCw is carried by the pair of points
w and −w and hence not by the Cˆ-atom of w for these w for which
ˆ
λCw = 12 (δw + δ−w ). Hence, the condition (iii) of proposition (3, § 7, 52)
is violated.
Let us now state and prove some simple, but useful consequences of
the previous propositions.
Let (Z, z) be a measurable space such that z is countably separating
and let ∀ z ∈ Z, the z-atom of z be z itself. This means that there exists
a sequence (Zn )n∈N of sets belonging to z such that every point z ∈ Z is
the intersection of a suitable subsequence of these Zn ’s.
Proposition 54. Let h be a mapping from Ω to Z such that h ∈ C . Let
(λCw )w∈Ω be a disintegration of λ with respect to C . Then,
w′ , h(w′ ) = h(w).
∀λ w, ∀λC
w
66
i.e. ∀λ w, h is λCw almost everywhere is a constant equal to h(w).
Proof. Let w ∈ Ω. There exists a subsequence (Znk )nk ∈N of (Zn )n∈N such
that
∞
\
h(w) =
Znk .
nk =1
Hence,
h−1 (h(w)) =
∞
\
h−1 (Znk ).
nk =1
h−1 (Z
∀ n ∈ N,
n ) ∈ C since h ∈ C .
Now, by condition (iii) of the proposition (3, § 7, 49), ∀ n ∈ N, ∀λ w,
λCw is carried by h−1 (Zn ) if w ∈ h−1 (Zn ). Hence, ∀λ w, ∀ n ∈ N, λCw is
carried by h−1 (Zn ) if w ∈ h−1 (Zn ).
Therefore, ∀λ w, λCw is carried by the intersection of all those h−1 (Zn )
for which w ∈ h−1 (Zn ). But the intersection of all the h−1 (Zn ) for which
wh−1 (Zn ) is h−1 (h(w)).
Hence, ∀λ w, λCw is carried by h−1 (h(w)). This precisely means that
w′ , h(w′ ) = h(w).
∀λ w, ∀λC
w
7. Conditions for a given family...
67
Corollary 55. Let (hn )n∈N be a sequence of functions on Ω with values
in Z such that ∀ n, hn ∈ C . Then,
w′ , ∀n ∈ N, hn (w′ ) = hn (w).
∀λ w, ∀λC
w
Proof. This is an immediate consequence of the above proposition (3, §
7, 54).
w′ , λCw′ =
Corollary 56. Let O be countably generated. Then ∀λ w, ∀λC
w
λCw .
Proof. Let (Bn )n∈N be a π-system generating O. Define a sequence
(hn )n∈N of functions on O as hn (w) = λCw (Bn )∀w ∈ Ω.
Then, ∀ n, hn is a function on Ω with values in the extended real 67
numbers and hn ∈ C ∀ n ∈ N. Hence, by the above Corollary (3, § 7,
55),
w′ , ∀n ∈ N, λCw′ (Bn ) = λCw (Bn ).
∀λ w, ∀λC
w
A standard application of the Monotone class theorem therefore
gives,
w′ , ∀B ∈ O, λCw′ (B) = λCw (B).
∀λ w, ∀λC
w
Hence,
w′ , λCw′ = λCw .
∀λ w, ∀λC
w
Part II
Measure valued
Supermartingales and
Regular Disintegration of
measures with respect to a
family of σ-Algebras
69
Chapter 4
Supermartingales
1 Extended real valued Supermartingales
Throughout this chapter, let (Ω, O, λ) be a measure space. Let (C t )t∈R 69
be a family of sub σ-algebras of Ôλ , which is increasing in the sense
that ∀ s, t ∈ R, s ≤ t, C s ⊂ C t . Let us further assume that λ restricted
to C t is σ-finite ∀t ∈ R. It C is a σ-algebra contained in Ôλ and f is a
function on Ω with values in R̄+ , belonging to Ôλ , f C will stand for a
conditional expectation of f with respect to C .
If f is a function from Ω × R to R̄+ , f t will denote ∀ t ∈ R, the
function on Ω with values in R̄+ , taking w ∈ Ω to f (w, t). ∀w ∈ Ω, fw
will denote the function on R to R̄+ , taking t to f (w, t).
Let f be a function from Ω × R to R̄+ . We have the following series
of definitions.
Definition 57. f is said to be adapted to (C t )t∈R if ∀ t, f t ∈ C t .
Definition 58. f is said to be right continuous if ∀λ w, fw is a right
continuous function.
Definition 59. f is said to be regulated if ∀λ w, fw has finite right and
finite left limits at all points t ∈ R.
Definition 60. A function g from Ω×R to R̄+ is said to be a modification
or a version of f if ∀ t ∈ R, ∀λ w, g(w, t) = f (w, t).
71
4. Supermartingales
72
Definition 61. f is said to be a supermartingale (resp. martingale)
adapted to (C t )t∈R if
70
(i) f is adapted to (C t )t∈R and
s
(ii) ∀ s, t ∈ R, s ≤ t, ∀λ w, ( f t )C (w) ≤ f s (w).
(resp.
(i) f is adapted to (C t )t∈R and
s
(ii) ∀ s, t ∈ R, s ≤ t, ∀λ w, ( f t )C (w) = f s (w))
Note that every martingale is a supermartingale. A simple example
of a supermartingale adapted to (C t )t∈R is given by any function f from
Ω × R to R̄+ , adapted to (C t )t∈R for which ∀λ w, fw is a decreasing
function of t.
Let g be any function on Ω with values in R̄+ and belonging to Ôλ .
Let ∀ t, w → f (w, t) be a given conditional expectation of g with respect
to C t . Then, it is clear that the function f on Ω × R to R̄+ , taking (w, t)
to f (w, t) is a martingale, adapted to (C t )t∈R .
t
t
R If f is a supermartingale adapted to (C )t∈R , note that t → J =
f t dλ is a decreasing function of t. If f is a right continuous supermartingale, we easily see by applying Fatou’s lemma that t → J t is right
continuous.
Definition 62. A supermartingale adapted to (C t )t∈R , is said to be regular if it is right continuous and regulated.
Note that if f is a regular supermartingale adapted to (C t )t∈R , then
∀λ w, fw is a real valued function on R.
Definition 63. A supermartingale g adapted to (C t )t∈R is said to be a
regular modification of a supermartingale f adapted to (C t )t∈R if g is
regular and is a modification of f .
Remark 64. If f is a supermartingale adapted to (C t )t∈R and if g1 and g2
are two regular modifications of f , we easily see that ∀λ w, ∀t, g1 (w, t) =
1. Extended real valued Supermartingales
71
73
g2 (w, t). In this sense, we say that a regular modification of a supermartingale adapted to (C t )t∈R is unique if exists. In particular, if f is a
regular supermartingale adapted to (C t )t∈R and if g is a regular modification of f , then ∀λ w, ∀t, g(w, t) = f (w, t).
The following theorem is very fundamental in the theory of supermartingales. It guarantees the existence of a regular modification for
a supermartingale, under some conditions. Whenever, we refer to the
following theorem, we refer to it as ‘the fundamental theorem’.
The Fundamental theorem. Let f be a supermartingale adapted to
(C t )t∈R such that ∀ t ∈ R, J t < +∞ and t → J t right continuous. Let
further the family (C t )t∈R of σ-algebras be right continuous in the sense
T u
that ∀t ∈ R, C t =
C . Then, there exists a function g on Ω × R with
u>t
values in R+ such that g is a regular modification of f .
This theorem is proved in P.A. Meyer [1] in theorems T 4 and T 3
of Chap. VI, in pages 95 and 94. In that book, the supermartingales
are assumed to take only real values. Hence to deduce the fundamental
theorem from the theorems T 4 and T 3 mentioned above, we observe the
following.
Since ∀ t, J t < +∞, we have ∀ t, ∀λ w, f (w, t) < +∞. Define a
function h on Ω × R with values in R+ as follows:



 f (w, t) if f (w, t) < +∞.
h(w, t) = 

0
otherwise.
Then, h is a modification of f with values in R+ . Hence by applying
the theorems T 4 and T 3 of Chap. VI of P.A. Meyer [1], we get a regular
modification for h and it is also a regular modification of f as well.
Remark 65. Let (C t )t∈R be an increasing right continuous family of σalgebras contained in Ôλ . Let f be a right Rcontinuous supermartingale
adapted to (C t )t∈R , such that ∀ t ∈ R, J t = f (w, t)dλ(w) < +∞. Then
f is regular.
For, by the fundamental theorem, there exists a regular modification
g of f since J t < +∞ ∀ t ∈ R and t → J t is right continuous by Fatou’s
4. Supermartingales
74
lemma.
∀t, ∀λ w, g(w, t) = f (w, t)
∀λ w, ∀t ∈ Q, g(w, t) = f (w, t).
72
Because ∀λ w, gw and fw are right continuous, it follows that ∀λ w,
∀t ∈ R, g(w, t) = f (w, t). Since ∀λ w, gw is regular, it follows that ∀λ w,
fw is also regular.
The following procedure is standard and gives a method of finding
out the regular
! modification of a supermartingale, whenever it exists.
k
Let n
be the set of all dyadic rationals. Then, ∀n ∈ N,
2 k∈Z,n∈N
S k k+1
R =
( n , n ]. ∀n ∈ N, define the functions τn from R to R as
2
k∈Z 2
follows:
k+1
k k+1
If t ∈ R, τn (t) = n if t ∈ ( n , n ].
2
2
2
Then ∀t ∈ R, τn (t) ↓ t as n → ∞ and if t is a dyadic rational, τn (t) = t
for all n large enough.
Let f be a function on Ω × R with values in R̄+ . Define another
function f¯ on Ω × R with values in R̄+ as follows:



 lim f τn (t) (w),
n→∞
¯f (w, t) = 



0,
if the limit exists and is finite.
otherwise.
It is obvious that if t is a dyadic rational and if f (w, t) < +∞, then
f¯(w, t) = f (w, t).
Proposition 66. Let f be a function on Ω × R with values in R̄+ . Then
(i) If the family (C t )t∈R is right continuous, and if f is adapted to
(C t )t∈R , so is f¯.
(ii) If f is right continuous, and if ∀λ w, ∀t, f (w, t) < +∞, then ∀λ w,
∀t, f¯(w, t) = f (w, t).
1. Extended real valued Supermartingales
Proof.
73
75
(i) Since f is adapted to (C t )t∈R and (C t )t∈R is right continuous, we can easily see that ∀ t, the set {w : lim f τn (t) (w) exists
n→∞
and is finite } belongs to τt . From this, it follows that ∀t, f¯t ∈ C
and this means that f¯ is adapted to (C t )t∈R .
(ii) Since f is right continuous, there exists a set N1 ∈ O with λ(N1 ) =
0 such that if w < N1 , fw is a right continuous function on R. Let
N2 ∈ O, λ(N2 ) = 0 be such that if w < N2 , f (w, t) < +∞ for
all t ∈ R. Then if w < N1 ∪ N2 , for all t, lim f τn (t) (w) exists
n→∞
and is equal to f (w, t) and hence is finite. Thus, if w < N1 ∪ N2 ,
f¯(w, t) = f (w, t)∀t ∈ R.
Hence, ∀λ w, ∀t ∈ R, f¯(w, t) = f (w, t).
We have the following obvious corollary.
Corollary 67. Let (C t )t∈R be a right continuous increasing family of sub
σ-algebras of Ôλ . Let f be a supermartingale adapted to (C t )t∈R and
let g be a regular modification of f . Then, ∀λ w, ∀t, g(w, t) = f¯(w, mt).
In particular, if f is a supermartingale adapted to (C t )t∈R with J t <
+∞ and t → J t right continuous, then ∀λ w, ∀t, lim f (w, τn (t)) exists
n→∞
and is finite and ∀ t, ∀λ w, f (w, t) = f¯(w, t). Moreover, f¯ is a regular
modification of f .
In the course of proofs of several theorems in this book, the following important theorem will be needed. We call it the “Upper envelope
theorem” and whenever we refer to it, we will refer to it as the “Upper
envelope theorem”. We state this without proof.
Let (X, X, µ) be a measure space. Let (Xt )t∈R be an increasing right
continuous family of σ-algebras contained in X̂µ . Let ( fn )n∈N be an increasing sequence of regular supermartingales in the sense that ∀ n ∈ N,
fn is a regular super martingale adapted to (Xt )t∈R such that ∀µ x, ∀t,
fn (x, t) ≤ fn+1 (x, t) for all n ∈ N. Let f (x, t) = sup fn (x, t). Then f is
n
a right continuous supermartingale and ∀µ x, limits from the left exist at
all points.
For a proof see P.A. Meyer [1]. T16, p.99.
4. Supermartingales
76
2 Measure valued Supermartingales
74
Let (Y, Y ) be a measurable space. Let ν be a measure valued function
on Ω × R with values in m+ (Y, Y ), taking a point (w, t) of Ω × R to the
measure νtw on Y . Let ∀ t ∈ R, νt be the measure valued function on
Ω with values in m+ (Y, Y ) taking w ∈ Ω to νtw and ∀ w ∈ Ω, let νw be
the measure valued function on R taking t ∈ R to the measure νtw . If f
is a function on Y, f ≥ 0, f ∈ Y , let ν( f ) be the function on Ω × R with
values in R̄+ taking (w, t) to νtw ( f ), ∀w ∈ Ω, let νw ( f ) be the function on
R taking t to νtw ( f ) and ∀ t ∈ R, let νt ( f ) be the function on Ω taking
w ∈ Ω to νtw ( f ).
Definition 68. ν is said to be a measure valued supermartingale (resp.
measure valued martingale) adapted to (C t )t∈R if
(i) ∀ t, νt ∈ C t
(ii) ∀ s, t ∈ R, s ≤ t and ∀ A ∈ C s and
Z
Z
νtw dλ(w) ≤
νws dλ(w)
A
A
(resp.
(i) ∀ t, νt ∈ C t
(ii) ∀ s, t ∈ R, s ≤ t and ∀ A ∈ C s ,
Z
Z
t
νw dλ(w) =
νws dλ(w) )
A
A
in the sense that ∀ function f on Y, f ∈ Y , f ≥ 0,
Z
Z
t
νw ( f )dλ(w) νws ( f )dλ(w)
(resp.
A
Z
A
A
νtw ( f )dλ(w)
=
Z
A
νws ( f )dλ(w). )
2. Measure valued Supermartingales
77
It is clear that ν is a measure valued supermartingale (resp. measure
valued martingale) adapted to (C t )t∈R if and only if ∀ function f on Y,
f ∈ Y , f ≥ 0, ν( f ) is and extended real valued supermartingale (resp.
martingale) adapted to (C t )t∈R .
Definition 69. ν is said to be a regular measure valued supermartin- 75
gale adapted to (C t )t∈R if ∀ B ∈ Y , ν(χB ) is a regular supermartingale
adapted to (C t )t∈R .
Definition 70. If ν and µ are two measure valued supermartingales,
adapted to (C t )t∈R , with values in m+ (Y, Y ), µ is said to be a modification of ν if ∀t, ∀λ w, µt = νt .
Definition 71. If ν and µ are two measure valued supermartingales with
values in m+ (Y, Y ) and adapted to (C t )t∈R , µ is said to be a regular
modification of ν if µ is regular and is a modification of ν.
Let ν be a measure valued supermartingale
adapted to (C t )t∈R , with
R
values in m+ (Y, Y ). Let ∀ t ∈ R, J t = νtw dλ(w). Then, ∀ t, J t is a
positive measure on Y . Since ν is a measure valued supermartingale,
∀ s, t ∈ R, s ≤ t, we have J t ≤ J s in the sense that ∀ B ∈ Y , J t (B) ≤
J s (B). In this sense we say that t → J t is decreasing. Let sup J t be
t∈R
the set function defined on Y as, ∀ B ∈ Y , (sup J t )(B) = sup J t (B).
t∈R
t∈R
With a similar definition for the set functions, sup J t , sup J t , we have
t∈Z
t∈Z
t≤0
sup J t = sup J t = sup since t → J t is decreasing. sup J t is a measure on
t∈R
t∈Z
t∈Z
t≤0
t∈Z
t≤0
Y , since it is the supremum of an increasing sequence of measures on
Y . Let us denote this measure by J. Thus,
J = sup J t = sup J t = sup J t .
t∈R
t∈Z
t∈Z
t≤0
78
4. Supermartingales
3 Properties of regular measure valued
supermartingales
In this section, let ν be a regular measure valued supermartingale with
values in m+ (Y, Y ), adapted to an increasing rightR continuous family
(C t )t∈R of σ-algebras contained in Ôλ . Let J t = νtw dλ(w) and J =
sup J t .
t∈R
76
Proposition 72. Let f be a function on Y, f ≥ 0, f ∈ Y such that f
is J-integrable. Then, ν( f ) is a regular supermartingale with values in
R̄+ , adapted to (C t )t∈R and ∀λ w, ∀t, f is νtw -integrable.
Proof. We know that ∀ B ∈ Y , ν(χB ) is a regular supermartingale, with
values in R̄+ adapted to (C t )t∈R . Hence if s is a step function on Y,
s ≥ 0, s ∈ Y , ν(s) is again a regular supermartingale with values in
R̄+ and adapted to (C t )t∈R . Since f is J-integrable, f ∈ Y and f ≥ 0,
there exists an increasing sequence (sn )n∈N of step functions such that ∀
n ∈ N, 0 ≤ sn ≤ f , sn ∈ Y and sn (y) ↑ f (y) for all y ∈ Y.
Hence ∀ t ∈ R, ∀ w ∈ Ω, νtw (sn ) ↑ νtw ( f ). Hence (ν(sn ))n∈N is an
increasing sequence of supermartingales with values in R̄+ , adapted to
(C t )t∈R , with limit as ν( f ). Hence, by the ‘Upper
R envelope theorem’.
ν( f ) is a right continuous supermartingale. ∀ t, νtw ( f )dλ(w) = J t ( f ) ≤
J( f ) < +∞. Hence by remark (4, § 1, 65), ν( f ) is a regular supermartingale. Hence ∀λ w, ∀t, νtw ( f ) < +∞ and this proves that ∀λ w, ∀t, f is
νtw -integrable.
Corollary 73. Let f be a function on Y, with values in R̄, f ∈ Y and
J-integrable. Then, ∀λ w, ∀t, f is νtw -integrable and ∀λ w, the function
νw ( f ) on R taking t to νtw ( f ) is regulated and right continuous.
Proof. This follows immediately from the above proposition (4, § 3, 72)
by writing f as f + − f − with the usual notation.
Proposition 74. Let E be a Banach space over R. Let g be a step function on Y with values in E, g ∈ Y and J-integrable. Then, ∀λ w, ∀t,
g is νtw -integrable and ∀λ w, νw (g), the function on R taking t to νtw (g),
is a right continuous and regulated function on R with values in E. (A
3. Properties of regular measure valued supermartingales
79
Banach space valued function f on R is said to be regulated if ∀ t ∈ R,
lim f (s) and lim f(s) exist in the Banach space).
s→t
s>t
s→t
s<t
Proof. Let g =
n
P
i=1
χAi xi where ∀ i, i = 1, . . . n, xi ∈ E, Ai ∈ Y and 77
Ai ∩ A j = ∅ if i , j. Since g is J-integrable, ∀ i = 1, . . . n, J(Ai ) <
+∞. Hence, by proposition (4, § 3, 72), ∀i = 1, . . . n, ν(Ai ) is a regular
supermartingale and hence ∀λ w, ∀t, i = 1, . . . , n, νtw (Ai ) < +∞ and ∀λ w,
∀i = 1, . . . n, νw (Ai ) is a right continuous regulated function on R.
n
P
Hence ∀λ w, ∀t, g is νtw -integrable and νtw (g) =
νtw (Ai )xi for these
i=1
w and for all t for which g is νtw -integrable.
P
Thus, ∀λ w, νw (g) = νw (Ai )xi on R. Hence ∀λ w, νw (g) is a right
continuous regulated function on R.
Proposition 75. Let E be a Banach space over R and let f be a function
on Y with values in E, f ∈ Y and J-integrable. Then, ∀λ w, ∀t, f is νtw integrable and ∀λ w, νw ( f ) is a right continuous regulated function on R
with values in E.
Proof. Since f is J-integrable, and ∈ Y , ∀n, ∃ a step function gn on Y,
R
1
gn ∈ Y such that |gn − f |dJ ≤ 2n .
2
∞
P
n
Let h =
2 |gn − f |.
n=1
Then h is ≥ 0 and is J-integrable on Y.
∀ n ∈ N, | f | ≤ |gn − f | + |gn | ≤
h
+ |gn |
2n
By proposition (4, § 3, 72), ∀λ w, ∀t, h is νtw -integrable and by proposition (4, § 3, 74) ∀λ w, ∀ n ∈ R, gn is νtw -integrable. Hence, ∀λ w, ∀t, f is
νtw -integrable.
1
|gn − f | ≤ n h.
2
1
Hence ∀λ w, ∀t, |νtw (gn ) − νtw ( f )| ≤ n νtw (h). Since h is ≥ 0 and J2
integrable, by proposition (4, § 3, 72), ∀λ w, νw (h) is a right continuous
80
4. Supermartingales
regulated function on R. Hence ∀λ w, νw (h) is locally bounded on R. 78
Hence ∀λ w, νtw (gn ) converges to νtw ( f ) in E, as n → ∞, locally uniformly
in the variable t.
By the previous proposition (4, § 3, 74), ∀λ w, ∀n ∈ N, νw (gn ) is right
continuous and regulated. Hence, since E is complete, and since ∀λ w,
the convergence of νtw (gn ) to νtw ( f ) is locally uniform in t, it follows that
νw ( f ) is also right continuous and regulated.
Chapter 5
Existence and Uniqueness of
regular modifications of
measure valued
supermartingales
1 Uniqueness theorem
Throughout this section, let (Ω, O, λ) be a measure space. Let (Y, Y ) 79
be a measurable space. Let ν be a measure valued supermartingale on
Ω × R with values in m+ (Y, Y ) adapted to an increasing right continuous
family (C t )t∈R of sub σ-algebras of Ôλ R. Let ν take a point (w, t) of Ω × R
to the measure νtw on Y . Let ∀ t, J t = νtw dλ(w) and let J = sup J T .
t∈R
Theorem 76 (Uniqueness). Let J be a finite measure on Y and let Y
have the J-countability property. If χ and Ψ are two regular modifications of ν, then ∀λ w, ∀t, χtw = Ψtw .
Proof. Since χ and Ψ are regular, χ(1) and Ψ(1) are regular supermartingales. Hence ∀λ w, ∀t, χtw (1) and Ψtw (1) are finite. Hence ∀λ w, ∀ t, χtw
and Ψtw are finite measures on Y . Thus, there exists a set N1 ∈ O with
λ(N1 ) = 0 such that if w < N1 , χtw and Ψtw are finite measures on Y for
81
82
5. Existence and Uniqueness of...
all t ∈ R.
Since Y has the J-countability property, there exists a set N ∈ Y
such that the σ-algebra Y ′ = Y ∩ ∁N on Y ′ = Y ∩ ∁N is countably
generated. Let B be a countable class generating Y ′ . Without loss of
generality, we can assume that B is a π-system, containing Y ′ .
Let B ∈ B. The supermartingales χ(B) and Ψ(B) are both regular
modifications of the supermartingale ν(B). Hence, by remark (4, § 1,
64),
∀λ w, ∀t, χtw (B) = Ψtw (B).
80
Thus, ∀ B ∈ B, ∀λ w, ∀ t, χtw (B) = Ψtw (B).
Since B is countable,
∀λ w, ∀B ∈ B, ∀ t, χtw (B) = Ψtw (B).
Hence there exists a set N2 ∈ O, with λ(N2 ) = 0 such that if w < N2 ,
∀t ∈ R, ∀B ∈ B, χtw (B) = Ψtw (B). Let w ∈ Ω. Consider the class Cw of
all sets C ∈ Y ′ such that ∀t, χtw (C) = Ψtw (C). If w < N1 ∪ N2 , the class
Cw is a d-system containing the π-system B. Hence, by the Monotone
class theorem, for w < N1 ∪ N2 , Cw contains the σ-algebra generated by
B which is Y ′ . Thus, if w < N1 ∪ N2 , ∀A ∈ Y ′ , ∀t, χtw (A) = Ψtw (A).
Now, since J(N) = 0, ∀t, J t (N) = 0.
Z
t
J (N) =
χtw (N)dλ(w).
Hence ∀t, ∀λ w, χtw (N) = 0. Therefore, ∀λ w, ∀t ∈ Q, χtw (N) = 0. Since
∀λ w, t → χtw (N) is right continuous, it follows that,
∀λ w, ∀t, χtw (N) = 0.
Similarly, ∀λ w, ∀t, Ψtw (N) = 0.
Hence, there exists a set N3 ∈ O with λ(N3 ) = 0 such that if w < N3 ,
∀ t, χtw (N) = Ψtw (N) = 0.
Let w < N1 ∪ N2 ∪ N3 . Let A ∈ Y .
A = A ∩ Y ′ ∪ A ∩ N.
2. Existence theorem
If
83
t ∈ R, χtw (A) = χtw (A ∩ Y ′ )
= Ψtw (A ∩ Y ′ )
= Ψtw (A).
Hence, if w < N1 ∪ N2 ∪ N3 , ∀t ∈ R, ∀A ∈ Y , χtw (A) = Ψtw (A).
Therefore,
∀λ w, ∀t, χtw = Ψtw on Y .
2 Existence theorem
Throughout this section, let us assume that (Ω, O, λ) is a measure space, 81
(C t )t∈R is an increasing right continuous family of σ-algebras contained
in Ôλ , Y is a topological space, Y is its Borel σ-algebra, ν is a measure
valued supermartingale
on Ω × R with values in m+ (Y, Y ), adapted to
R
t
t
t
(C )t∈R , J = νw dλ(w) and J = sup J t .
t∈R
Definition 77. We say t → J t is right continuous if ∀ function f on Y,
f ≥ 0, f ∈ Y , t → J t ( f ) is right continuous.
Theorem 78 (Existence). Let J be a finite measure on Y and let t → J t
be right continuous. Let Y have the J-compacity metrizability property.
Then, there exists a regular modification of ν.
Proof. Since J is a finite measure, ∀t, J t is also a finite measure. Hence
∀t, ∀λ w, νtw is a finite measure. Define the measure valued function ν′
on Ω × R as



νtw , if νtw is a finite measure
′t
νw=

0,
otherwise.
Then, ν′ is a modification of ν and for all w ∈ Ω, for all t ∈ R, ν′ tw is
a finite measure on Y . ν′ , being a modification of ν, has the same J t ∀t
and the same J. Hence, if we can prove the theorem for ν′ under the
assumptions mentioned in the statement of the theorem, the theorem for
5. Existence and Uniqueness of...
84
82
ν is obvious. Hence, without loss of generality, we shall assume that ∀t,
∀w, νtw is a finite measure on Y .
Let us divide the proof in two cases Case I and Case II. In Case I, we
shall prove the theorem assuming Y to be a compact metrizable space
and in case II, we shall deal with the general case and deduce the result
from Case I.
In Case I the proof is carried out in four steps, Step I, Step II, Step
III and Step IV.
In Step I, we define a measure valued function ν̃ on Ω×R with values
in + (Y, Y ) such that ∀t ∈ R and ∀w ∈ Ω, ν̃tw is a Radon measure on Y.
In Step II, we prove that ∀t, ν̃t ∈ C t .
In Step III, we prove that if C (Y) is the space of all real valued
continuous functions on Y, ∀ϕ ∈ C (Y), ∀λ w, the function ν̃w (ϕ) on R
taking t ∈ R to ν̃tw (ϕ), is regular and ν̃ is a modification of ν.
In Step IV, we prove that ν̃ is a regular measure valued supermartingale.
Case I. Y, a compact metrizable space.
Since J, J t are finite measures on Y , they are Radon measures on Y,
since Y is a compact metrizable space. Let C (Y) be the vector space of
all real valued continuous functions on Y and C+ (Y), the cone of all the
positive real valued continuous functions on Y.
RStep I. Let ϕ ∈ C+ (Y). Consider the real valued supermartingale ν(ϕ).∀t,
νtw (ϕ)dλ(w) = J t (ϕ) is finite and by hypothesis, t → J t (ϕ) is right
continuous. Hence, by the fundamental theorem, a regular modification
for ν(ϕ) exists. Hence, by Corollary (4, § 1, 67), ∀λ w, ∀t, lim ντwn (t) (ϕ)
n→∞
exists and is finite, and if ν(ϕ)(w, t) = lim ντwn (t) (ϕ) if this limit exists
n→∞
and is finite, and = 0 otherwise, then, ν(ϕ) is a regular modification of
ν(ϕ).
Hence, if ϕ ∈ C (Y), ∀λ w, ∀t, lim ντwn (t) (ϕ) exists and is finite.
n→∞
Thus, for ϕ ∈ C (Y), if
Ω◦t,ϕ = {w ∈ Ω | lim ντwn (t) (ϕ) exists and is finite }.
n→∞
and if Ω◦ϕ =
T
t∈R
Ω◦t,ϕ , then λ is carried by Ω◦ϕ and hence a priori by Ω◦t,ϕ ,
2. Existence theorem
85
∀t ∈ R. Also, ∀ϕ ∈ C (Y), ∀t ∈ R, Ω◦t,ϕ ∈ C t .
Let D be a countable dense subset of C (Y), containing 1. Let Ω◦ = 83
T ◦
T ◦
T ◦
Ωt .
Ωt,ϕ . Note that Ω◦ =
Ωϕ and Ω◦t =
t∈R
ϕ∈D
ϕ∈D
Since D is countable, λ is carried by Ω◦ , Ω◦t ∈ C t ∀t ∈ R and λ is
carried by Ω◦t for every t ∈ R.
Let w ∈ Ω◦t . Since 1 ∈ D, lim ντwn (t) (1) exists and is finite. Hence ∀t,
n→∞
sup ντwn (t) (1) is finite.
(1)
n
Also if w ∈ Ω◦t , ∀ϕ ∈ D, lim ννwn (t) (ϕ) exists and is finite.
(2)
Ω◦t ,
then ∀ϕ ∈
n→∞
From (1) and (2), we can easily deduce that if w ∈
C (Y), lim ντwn (t) (ϕ) exists and is finite. Thus,
n→∞
Ω◦t =
\
Ω◦t,ϕ .
\
Ω◦ϕ .
ϕ∈C (Y)
In the same way,
Ω◦ =
ϕ∈C (Y ′ )
Thus, if w ∈ Ω◦t , the vague limit of ντwn (t) exists. Conversely if for a
w ∈ Ω and for a t ∈ R, the vague limit of ντwn (t) exists, then it is easy to
see that w ∈ Ω◦t . Thus,
Ω◦t = {w ∈ Ω | vague limit of ντwn (t) exists }.
Define the measure valued function ν̃ on Ω × R with values in
m+ (Y, Y ) as follows.

τ (t)


vague limit of νwn , if w ∈ Ω◦t
ν̃tw = 

0, otherwise.
ν̃tw is thus a Radon measure on Y for all w ∈ Ω and for all t ∈ R.
Step II. Let show that ∀t, ν̃t ∈ C t .
For this, we have to show that ∀B ∈ Y , the function ν̃t (B) ∈ C t .
Let us first show that if ϕ ∈ C+ (Y), then ν̃t (ϕ) ∈ C t .
84
5. Existence and Uniqueness of...
86



lim ντwn (t) (ϕ), if w ∈ Ω◦t

n→∞
ν̃tw (ϕ) = 


0, otherwise.
Thus, ν̃tw (ϕ) = χΩ◦t
w→
lim supντwn (t) (ϕ)
n→∞
lim supντwn (t) (ϕ). Since Ω◦t ∈ C t and since
n→∞
belongs to C t , it follows that ν̃t (ϕ) ∈ C t .
Let U be an open subset of Y, U , ∅. Then, since Y is metrizable,
there exists an increasing; sequence (ϕn )n∈N of continuous functions on
Y such that 0 ≤ ϕn ≤ χn ∀ n ∈ N and ϕn (y) ↑ χU (y) for all y ∈ Y.
Hence, ∀ t ∈ R, ∀w ∈ Ω, ν̃tw (χU ) = lim ν̃tw (ϕn ). Since ∀ n ∈ N,
n→∞
∀ t ∈ R, ν̃t (ϕn ) ∈ C t , it follows that ∀t ∈ R, ν̃t (χU ) also belongs to C t .
Now, the standard application of the Monotone class theorem gives
that ∀ B ∈ Y , ∀t ∈ R, ν̃t (χB ) ∈ C t . Hence ∀ t ∈ R, ν̃t ∈ C t .
Step III. Let us shown that ∀ ϕ ∈ C (Y), ∀λ w, ν̃w (ϕ) is regular and that
ν̃ is a modification of ν.
Let w ∈ Ω◦ . Then ∀ ϕ ∈ C (Y), ∀t, lim ννwn (t) (ϕ) exists and is finite
n→∞
and this limit is equal to ν̃tw (ϕ). Therefore, if ϕ ∈ C+ (Y) and w ∈ Ω◦ ,
ν̃tw (ϕ) = ν(ϕ)(w, t)
for all t. Since λ is carried by Ω◦ , we therefore have
∀λ w, ∀t, ∀ϕ ∈ C+ (Y), ν̃tw (ϕ) = ν(ϕ)(w, t).
(1)
Now, ν(ϕ) is a regular modification of ν(ϕ) by Corollary (4, § 1, 67).
Hence, ∀λ w, ν(ϕ)w is regular. Therefore, since from (1).
∀λ w, ∀ t, ∀ϕ ∈ C+ (Y), ν̃tw (ϕ) = ν(ϕ)(w, t),
85
it follows that, ∀ϕ ∈ C+ (Y), ∀λ w, ν̃w (ϕ) is regular. Hence ∀ϕ ∈ C (Y),
∀λ w, ν̃w (ϕ) is also regular.
If ϕ ∈ C+ (Y), since ν(ϕ) is a modification of ν(ϕ), we have
∀ϕ ∈ C+ (Y),
∀λ w, ν(ϕ) (w, t) = νtw (ϕ).
Hence from this and from (1), we deduce that
∀ϕ ∈ C+ (Y), ∀t,
∀λ w, ν̃tw(ϕ) = νtw (ϕ).
2. Existence theorem
87
Hence ∀ϕ ∈ C (Y), ∀t, ∀λ w, ν̃tw (ϕ) = νtw (ϕ). Therefore, since D is
countable,
∀t, ∀λ w, ν̃tw (ϕ) = νtw (ϕ)
for all ϕ ∈ D.
Since D is dense and the measures ν̃tw and νtw are finite measures for
all w ∈ Ω and for all t ∈ R, it follows that
∀t,
∀λ w,
∀ϕ ∈ C (Y), ν̃tw (ϕ) = νtw (ϕ)
and hence, ∀t, ∀λ w, ν̃tw = νtw as νtw and ν̃tw are Radon measures for all
w ∈ Ω and for all t ∈ R.
Hence ν̃ is a modification of ν.
Step IV.
We shall show that ν̃ is regular. For this we have to show that ∀B ∈
Y , ∀λ w, ν̃w (χB ) is regular.
From Step III, if ϕ ∈ C+ (Y), ∀λ w, ν̃w (ϕ) is regular. In particular,
∀λ w, ν̃w (1) is regular. Hence, ∀λ w, ν̃w (1) is a locally bounded function
on R i.e. ∃ a set N ∈ O with λ(N) = 0 such that if w < N, ν̃w (1) is a
locally bounded function on R. Hence if B ∈ Y and w < N, ν̃w (χB ) is
also a locally bounded function on R.
Let U be an open set of Y, U , ∅. Then since Y is metrizable, there
exists an increasing sequence (ϕn )n∈N of continuous functions on Y such 86
that ∀n ∈ N, 0 ≤ ϕn ≤ χU and
ϕn (y) ↑ χU (y) for all y ∈ Y.
Hence ∀ w ∈ Ω, ∀ t ∈ R, ν̃tw (ϕn ) ↑ ν̃tw (χU ). Since ∀n, ν̃(ϕn ) is a regular
supermartingale and since ν̃(ϕn ) is an increasing sequence of functions
on Ω×R, it follows from the ‘Upper envelope theorem’ that ∀λ w, ν̃w (χU )
is right continuous and has limits form the left at all points t ∈ R. The
finiteness of these limits from the left at all t ∈ R, follows from the fact
that ∀λ w, ν̃w (χU ) is locally bounded. Hence, ∀λ w, ν̃w (χU ) is regular.
Let C = {C ∈ Y | ∀λ w, ν̃w(χC ) is regular }. Then, the class C
is a d-system again by the ‘Upper envelope theorem’ and by the local
boundedness of ν̃w (χB ) ∀B ∈ Y , for almost all w ∈ Ω. This class C
contains the class U of all open sets which is a π-system generating Y .
5. Existence and Uniqueness of...
88
Hence, C contains the σ-algebra Y and hence is equal to Y . Thus,
∀B ∈ Y , ∀λ w, ν̃w (χB ) is regular.
Since ν̃ is a modification of ν, ν̃(χB ) is a supermartingale ∀ B ∈ Y .
Moreover, since ∀ B ∈ Y , ∀λ w, ν̃w (χB ) is regular, ν̃(χB ) is a regular
supermartingale ∀B ∈ Y . Hence, ν̃ is a regular supermartingale and is
a regular modification of ν.
Thus, the proof in case I is complete. Note that in this case, because
of the uniqueness theorem, the regular modification of ν is unique.
Case II. Y, a general topological space having the J-compactiy metrizability property.
Since Y has the J-compacity metrizability property, there exists a
sequence (Xn )n∈N of compact metrizable sets and a set N ∈ Y such that
S
Xn ∪ N.
J(N) = 0 and Y =
n∈N
87
Let ∀n ∈ N, Yn =
n
S
Xi . Then ∀n, Yn is again a compact metrizable
i=1
set. This is because ∀n, Yn is both compact and Suslin each Xi is so.
S
Yn ∪ N.
Moreover, the sequence (Yn )n∈N is increasing and Y =
n∈N
Let ∀n ∈ N, Yn be the Borel σ-algebra of Yn . Let ∀ n ∈ N, νn
be the measure valued function on Ω × R with values in m+ (Yn , Yn )
associating to each (w, t) ∈ Ω × R, the measure νn,t
w on Yn , which is the
restriction of the measure νtw to Yn . Then, it is easily seen that ∀ n, νn is
a measureRvalued supermartingale on Ω × R with values in m+ (Yn , Yn ).
t
t
Let Jnt = νn,t
w dλ(w) and Jn = sup Jn . Then, ∀ n ∈ N, Jn and Jn are
t∈R
respectively the restriction of J t and J to Yn . Hence ∀ n, ∀t, Jnt is finite,
Jn is finite and t → Jnt is right continuous. Hence the hypothesis of
the theorem is verified for νn ∀n ∈ N. Hence by Case I, ∀n ∈ N, ∃
a unique measure valued supermartingale ν̃n with values in m+ (Yn , Yn )
n
taking (w, t) to ν̃n,t
w , which is a regular modification of ν .
n+1,t Let ∀ n ∈ N, ∀t, ν̃w Yn denote the restriction of the measure ν̃n+1,t
w
to Yn and let ν̃n+1
Yn denote the measure valued function on Ω × R with
| Yn . Then it is easily
values in m+ (Yn , Yn ) associating (w, t) to ν̃n+1
w
n+1
seen that ∀n, ν̃Yn is also a regular modification of νn . Hence, because
of uniqueness,
n+1,t
∀λ w, ∀t, ν̃n,t
| Yn
(1)
w = ν̃w
2. Existence theorem
89
∀ n ∈ N, let us consider the measures ν̃n,t
w as measures on Y by defining
it to be zero for any set B ∈ Y for which B ∩ Yn = ∅.
Then from (1), it follows that ∀λ w, ν̃n,t
w is an increasing sequence
of measures on Y for all t ∈ R. i.e. there exists a set N1 ∈ O with
λ(N1 ) = 0 such that if w < N1 , for all t ∈ R, (ν̃n,t
w )n∈N is an increasing
sequence of measures on Y .
Now, ∀ n ∈ N, define the measure valued function µn on Ω × R, with
values in m+ (Y, Y ) taking (w, t) to µn,t
w as follows.

n,t


ν̃w , if w < N1
n,t
µw = 

0,
if w ∈ N1
Then ∀w ∈ Ω, ∀t ∈ R, (µn,t
w )n∈N is an increasing sequence of measures 88
of Y . ∀n ∈ N, µn is a regular measure valued supermartingale on Ω × R
with values in m+ (y, Y ).
Define ∀w ∈ Ω, and ∀t ∈ R, the measures µtw as sup µn,t
w . Then
n
∀ B ∈ Y , µtw (B) = sup µn,t
w (B).
n
Let µ be the measure valued function on Ω × R taking (w, t) to µtw .
Since ∀B ∈ Y , ∀n ∈ N, µn (χB ) is a regular supermartingale, it
follows form the ‘Upper enveloppe theorem’ that ∀λ w, µw (χB ) is right
continuous and limits from the left exist at all t ∈ R. Hence µ(χB ) is a
right continuous supermartingale and ∀ t ∈ R,
Z
Z
µn,t
µtw (χB )dλ(w) = lim
w (χB )dλ(w)
n→∞
Z
ν̃n,t
= lim
w (χB )dλ(w)
n→∞
Z
νn,t
= lim
w (χB )dλ(w)
n→∞
Z
= lim
νtw (B ∩ Yn )dλ(w)
n→∞
= lim J t (B ∩ Yn )
n→∞
= J t (B) < +∞.
5. Existence and Uniqueness of...
90
89
Hence by remark (4, § 1, 65), µ(χB ) is a regular supermartingale. Therefore µ is a regular measure valued supermartingale.
We have to show that µ is a modification of ν. i.e. we have to show
that ∀t, ∀λ w, µtw = νtw .
Let t ∈ R be fixed. Since J(N) = 0, J t (N) = 0. Hence ∀λ w, νtw (N) =
0 i.e. ∃ Nt1 ∈ O such that λ(Nt1 ) = 0 and if w < Nt1 , then νtw (N) = 0.
∀n ∈ N, ν̃n is a modification of νn . Hence,
∀ n ∈ N,
n,t
∀λ w, ν̃n,t
w = νw .
n,t
2
Therefore, ∀λ w, ∀n ∈ N, ν̃n,t
w = νw . Hence there exists a set Nt ∈ O
2
2
with λ(Nt ) = 0 such that if w < Nt ,
n,t
ν̃n,t
w = νw
for all n ∈ N,
Let M = Nt1 ∪ Nt2 ∪ N1 . Then M ∈ O and λ(M) = 0. If w < M, and
if B ∈ Y ,
νtw (B) = lim νtw (Yn ∩ B)
n→∞
= lim νn,t
w (B)
n→∞
= lim ν̃n,t
w (B)
n→∞
= lim µn,t
w (B)
n→∞
= µtw (B).
Hence, if w < M, νtw = µtw . Therefore
∀ t,
∀λ w, µtw = νtw .
This proves that µ is a modification of ν.
3 The measures νtw and the left limits
90
Throughout this section, let (Ω, O, λ) be a measure space. Let (C t )t∈R
be an increasing right continuous family of σ-algebras contained in Ôλ .
3. The measures νtw and the left limits
91
91
Let Y be a compact metrizable space and Y be its Borel σ-algebra. Let
ν be a regular measure valued supermartingale on Ω ×RR with values
in m+ (Y, Y ), adapted to (C t )t∈R . Let ∀ t ∈ R, J t = νtw dλ(w) and
J = sup J t . Let us assume that J is a finite measure on Y .
t∈R
Since ν is a regular measure valued supermartingale, ∀λ w, νw (1) is a
right continuous, regulated function on R. Hence ∀λ w, νw (1) is locally
bounded on R. i.e. ∃ a set N1 ∈ O such that λ(N1 ) = 0 and if w < N1 ,
νw (1) is locally bounded on R.
∀ϕ ∈ C (Y), the space of all real valued continuous functions on
Y, ∀λ w, νw (ϕ) is a right continuous regulated function on R. Hence,
∀ϕ ∈ C (Y), ∀λ w, t → νtw (ϕ) is right continuous and has finite left limits
at all points of R.
Let D be a countable dense subset of C (Y). We have,
∀λ w, ∀ϕ ∈ D, t → νtw (ϕ) is right continuous and has finite left limits
at all points of R. i.e. there exists a set N2 ∈ O such that if w < N2 ,
∀ϕ ∈ D, t → νtw (ϕ) is right continuous and has finite left limits at all
points of R.
Let w < N1 ∪ N2 . Then it is easy to see that ∀ϕ ∈ C (Y), t → νtw (ϕ)
is right continuous and has finite limits at all points of R.
Hence, define ∀ w ∈ Ω, ∀ t ∈ R, the linear mappings νt−
w on C (Y),
with values on R as follows, ∀ϕ ∈ C (Y),



lim ν s (ϕ), if w < N1 ∪ N2


 s→t w
t−
νw (ϕ) = 
s<t



0, if w ∈ N1 ∪ N2 .
Then, ∀ w ∈ Ω, ∀ t ∈ R, νt−
w is a positive linear functional on C (Y)
and thus defines a Radon measure on Y. And, by definition,
s
∀λ w, ∀t, νt−
w (ϕ) = lim νw (ϕ)
s→t
s<t
∀ ϕ ∈ C (Y).
Proposition 79. Let ( fn )n∈N be an increasing sequence of functions on
Yn , fn ≥ 0, fn ∈ Y and J-integrable ∀ n ∈ N, such that fn increases
everywhere to a J-integrable function f . Further, let ∀λ w, ∀ n, ∀t, 92
5. Existence and Uniqueness of...
92
s
t−
νt−
w ( fn ) = lim νw ( fn ). Then, ∀λ w, ∀ t, f is νw integrable and ∀λ w, ∀t,
νt−
w (f)
s→t
s<t
= lim νws ( f )
s→t
s<t
Proof. By passing to a subsequence if necessary, we shall assume that
∀ n ∈ N,
Z
1
| fn − f |dJ ≤ 2n .
2
∞
P n
Let h =
2 | fn − f |. Then h ≥ 0, h ∈ Y and J-integrable.
n=1
∀w, ∀t, |νtw ( fn ) − νtw ( f )| ≤
1 t
ν (h).
2n w
By proposition (4, § 3, 72), ∀λ w, νw (h) is a right continuous, regulated function on R and hence is locally bounded. Hence ∀λ w, νtw ( fn )
converges to νtw ( f ) as n → ∞ locally uniformly in the variable t. Hence,
∀λ w, ∀ t ∈ R, lim lim νws ( fn ) = lim νws ( f ).
n→∞ s→t
s<t
s→t
s<t
(Proposition (4, § 3, 72) guarantees the existence of lim νws ( fn ) and
lim νws ( f ),
s→t
s<t
s→t
s<t
∀ n, ∀t). Hence,
s
∀λ w, ∀t ∈ R, lim νt−
w ( fn ) = lim νw ( f ).
n→∞
s→t
s<t
Now, since f is J-integrable, by proposition (4, § 3, 72), ∀λ w, νw ( f )
is a right continuous regulated function on R and hence
∀λ w, ∀t, lim νws ( f ) < +∞.
s→t
s<t
Therefore, ∀λ w, ∀t ∈ R lim . But ∀w ∈ Ω, ∀t ∈ R, lim νt−
w ( fn ) =
n→∞
νt−
w ( f ) since fn ↑ f everywhere on Y. Hence,
∀λ w, ∀t ∈ R, νt−
w ( f ) < +∞
n→∞
3. The measures νtw and the left limits
93
and
s
∀λ w, ∀t ∈ R, νt−
w ( f ) = lim νw ( f ).
s→t
s<t
Theorem 80. ∀ B ∈ Y , ∀λ w, ∀t
s
νt−
w (χB ) = lim νw (χB ).
s→t
s<t
Proof. Note that ∀ B ∈ Y , ∀λ w, ∀t, lim νws (χB ) exists and is finite, since
s→t
s<t
ν(χB ) is a regular supermartingale.
Let U be an open set, U , ∅. Then χU is lower semi continuous and
J-integrable since J is a finite measure. Since Y is a metrizable space, ∃
an increasing sequence (ϕn )n∈N of real valued non-negative continuous
functions on Y such that ϕn (y) ↑ χU (y) for all y ∈ Y.
s
∀λ w, ∀t, ∀n, νt−
w (ϕn ) = lim νw (ϕn ).
s→t
s<t
Hence, by the previous proposition (5, § 3, 79), ∀λ w, ∀t, νt−
w (χU ) =
lim νws (χU ). Let
s→t
s<t
s
C = {C ∈ Y | νt−
w (χC ) = lim νw (χC )}.
s→t
s<t
This class is a d-system, again by the previous proposition (5, § 3,
79). This d-system contains the π-system U of all open subsets of Y.
Hence, C contains the σ-algebra Y , which is generated by U. Hence,
s
∀ B ∈ Y , νt−
w (χB ) = lim νw (χB ).
s→t
s<t
Proposition 81. Let f be any J-integrable extended real valued function
on Y, f ∈ Y . Then, ∀λ w, ∀t, f is νt−
w -integrable and
′
∀λ w, ∀t, νt−
νtw ( f ).
w ( f ) = lim
′
t →t
t′ <t
93
5. Existence and Uniqueness of...
94
′
Proof. Note that the existence of lim
νtw ( f ) ∀ t ∈ R is guaranteed by
′
t →t
t′ <t
Corollary (4, § 3, 73).
Sufficient to prove when f is ≥ 0, f ∈ Y and J-integrable.
If s is any step function on Y, 0 ≤ s ≤ f and s ∈ Y , then s is
J-integrable and it follows from the previous theorem (5, § 3, 80), that
′
∀λ w, ∀t ∈ R, νt−
νtw (s).
w (s) = lim
′
t →t
t′ <t
Now, we can find as increasing sequence (sn )n∈N of step functions
on Y, sn ∈ Y and 0 ≤ sn ≤ f ∀ n ∈ N such that ∀y ∈ Y, sn (y) ↑ f (y).
Now, from proposition (5, § 3, 79) it follows that ∀λ w, ∀t, f is νt−
wintegrable and
′
∀λ w, ∀ t, νt−
νtw ( f ).
w ( f ) = lim
′
t →t
t′ <t
Proposition 82. Let E be a Banach space over R. Let g be a step function on Y, with values in E, g ∈ Y and J-integrable. Then, ∀λ w, ∀t, g is
νt−
w -integrable and
′
∀λ w, ∀t, νt−
νtw (g).
w (g) = lim
′
t →t
t′ <t
′
Proof. Note that the existence of lim
νtw (g)∀t, is guaranteed by proposi′
t→
t′ <t
tion (4, § 3, 74).
Let g be of the form
n
P
i=1
94
χAi xi where ∀ i = 1, . . . , n, Ai ∈ Y , xi ∈ E
and Ai ∩ A j = ∅ if i , j.
Since ∀w ∈ Ω, ∀t ∈ R, νt−
w is a finite measure on Y , ∀i, ∀w ∈ Ω,
t−
∀t ∈ R, νw (Ai ) < +∞. Hence ∀ w ∈ Ω, ∀ ∈ R, g is νt−
w -integrable,
and ∀w ∈ Ω, ∀t ∈ R, νt−
(A
)
<
+∞.
Hence
∀
w
∈
Ω,
∀
t ∈ R, g is
i
w
νt−
-integrable,
and
∀w
∈
Ω,
∀t
∈
R,
w
νt−
w (g) =
n
X
i=1
νt−
w (Ai )xi .
3. The measures νtw and the left limits
95
By theorem (5, § 3, 80),
′
∀λ w, ∀t, ∀i, νt−
νtw (Ai ).
w (Ai ) = lim
′
t →t
t′ <t
Hence ∀λ w, ∀ t,
νt−
w (g) =
n
X
′
lim
νtw (Ai )xi
′
t →t
i=1 t′ <t
n
′
= lim
lim νtw (Ai )xi
′
=
t →t i=1
t′ <t
′
lim
νtw (g)
′
t →t
t′ <t
Theorem 83. Let f be a J-integrable function on Y, with values in a
Banach space E over R, f ∈ Y . Then ∀λ w, ∀t, f is νt−
w -integrable and
′
∀λ w, ∀t, νt−
νtw ( f ).
w ( f ) = lim
′
t →t
t′ <t
′
Proof. Note that the existence of lim
νtw ( f ), ∀λ w, for every t is guaran′
t →t
t′ <t
teed by proposition (4, § 3, 75).
There exists a sequence (gn )n∈N of step functions on Y with values
in E, gn ∈ Y ∀n ∈ N such that
Z
1
∀ n ∈ N,
|gn − f |dJ ≤ 2n .
2
∞
P n
Let h =
2 |gn − f |. Then h ≥ 0, h ∈ Y and is J-integrable.
95
n=1
h
+ |gn |.
2n
By proposition (5, § 3, 81), ∀λ w, ∀t, h is νt−
w -integrable, and by the
previous proposition (5, § 3, 82), ∀w, ∀t, ∀n, gn is νt−
w -integrable. Hence,
-integrable.
Moreover,
∀
w,
∀t,
∀λ w, ∀t, f is νt−
λ
w
∀n, | f | ≤ |gn − f | + |gn | ≤
t−
t−
| νt−
w (gn ) − νw ( f ) | =| νw (gn − f ) |
5. Existence and Uniqueness of...
96
≤
1 t−
ν (h).
2n w
t−
Since ∀λ w, ∀t, νt−
w (h) < +∞, it follows that lim νw (gn ) exists and is
n→∞
equal to νt−
w ( f ).
(1)
∀λ w, ∀t, |∀tw (gn ) − νtw ( f )| ≤ νtw (|gn − f |) ≤
1 t
ν (h).
2n w
By proposition (4, § 3, 72), ∀λ w, νw (h) is right continuous and regulated. Hence, ∀λ w, νw (h) is a locally bounded function on R.
Hence, ∀λ w, νtw (gn ) converges to νtw ( f ) in E as n → ∞ locally uniformly in the variable t. Hence
′
′
lim lim
νtw (gn ) = ′lim νtw ( f ).
′
n→∞ t →t
t′ <t
t →∞
t′ <t
Therefore,
′
lim νt−
νtw ( f )
w (gn ) = lim
′
n→∞
t →t
t′ <t
(2)
From (1) and (2), we see that
′
∀λ w, ∀t, νt−
νtw ( f ).
w ( f ) = lim
′
t →t
t′ <t
Chapter 6
Regular Disintegrations
1 Basic Definitions
Throughout this chapter, let us assume that (Ω, O, λ) is a measure space, 96
(C t )t∈R is an increasing right continuous family of sub σ-algebras of Ôλ ,
λ restricted to C t is σ-finite ∀ t ∈ R and that ∀t, λ has a disintegration
(λtw )w∈Ω with respect to C t .
Then (w, t) → λtw is a measure valued martingale on Ω × R with
values in m+ (Ω, O), adapted to (C t )t∈R .
Definition 84. (λtw )w∈Ω is said to be a regular disintegration of λ with respect to the family (C t )t∈R if (w, t) → λtw is a regular martingale adapted
to (C t )t∈R .
From the theorem (3, § 6, 44), we see that if Ω is a topological
space, O is its Borel σ-algebra and Ω has the λ-compacity metrizability property, then ∀ t ∈ R, λ has a disintegration (δtw )w∈Ω with respect
to C t · (w, t) → δtw is a measure valued martingale adapted to (C t )t∈R .
If further, λ is finite, then by theorem (5, § 2, 78), this measure valued
martingale has a regular modification, say (λtw )w∈Ω . Then, (λtw )w∈Ω is
t∈R
t∈R
a regular disintegration of λ with respect to (C t )t∈R . Hence, if we assume that Ω is a topological space, O is its Borel σ-algebra, Ω has the
λ-compacity metrizability property, (C t )t∈R is an increasing right continuous family of σ-algebras contained in Ôλ and λ is a finite measure
97
6. Regular Disintegrations
98
on O, then λ has a regular disintegration with respect to (C t )t∈R .
Definition 85. A set F ⊂ Ω×R is said to be λ-evanescent if its projection
on Ω is of λ-measure zero.
97
Note that if F is a λ-evanescent set, then there exists a set B ∈ Ôλ
such that λ(B) = 0 and F ⊂ B × R. Hence, if a property P(w, t) holds
except for a λ-evanescent set, then there exists a set B ∈ Ôλ with λ(B) =
0 such that if w < B, P(w, t) holds for all t ∈ R. Conversely, if there exists
a set B ∈ Ôλ with λ(B) = 0, such that a property P(w, t) holds for all t,
whenever w < B, then the property holds except for the λ-evanescent set
B × R.
2 Properties of regular disintegrations
Let us assume hereafter that (λtw )w∈R is a regular disintegration of λ with
respect to (C t )t∈R .
t∈R
Proposition 86. Let B ∈ O with λ(B) = 0. Then ∀λ w, ∀t, λtw (B) = 0.
Proof. We have ∀t, ∀λ w, λtw (B) = 0. Hence,
∀λ w, ∀t ∈ Q, λtw (B) = 0.
Since ∀λ w, t → λtw (B) is right continuous, it follows that
∀λ w, ∀t ∈ R, λtw (B) = 0.
Proposition 87. λtw is a probability measure except for a λ-evancescent
set.
Proof. ∀t, ∀λ w, λtw is a probability measure. Therefore, ∀λ w, ∀t ∈ Q,
λtw is a probability measure.
Since ∀λ w, t → λtw (Ω) is right continuous, it follows that ∀λ w, ∀t ∈
R, λtw is a probability measure.
2. Properties of regular disintegrations
99
Proposition 88. Let A ∈ C t ∀t ∈ R. Then ∀λ w, ∀t, λtw is carried by A or
by ∁A, according as w ∈ A or w ∈ ∁A.
98
Proof. By proposition (3, § 7, 49), we know that ∀t, ∀B ∈ C t , ∀λ wλtw is
carried by B or by ∁B according as w ∈ B or w ∈ ∁B. i.e. ∀ t, ∀ B ∈ C t ,
∀λ w, χB (w) · λtw (∁B) and χ∁B (w) · λtw (B) are both zero.
Since A ∈ C t for all t, we have therefore, ∀λ w, ∀t ∈ QχA (w)·λtw (∁A)
and χ∁A (w) · λtw (A) are both zero.
Since ∀λ w, t → λtw (∁A) and t → λtw (A) are right continuous, it
follows that ∀λ w, ∀t ∈ R, χB (w) · λtw (∁A) and χ∁A (w)λtw (A) are both
zero.
Proposition 89. Let f be a bounded regular supermartingale, adapted
to the family (C t )t∈R . Then, ∀λ w, ∀s, t → λwB ( f t ) is right continuous.
Proof. Since f is a regular supermartingale, ∀λ w, fw is right continuous,
i.e. ∃B ∈ O with λ(B) = 0 such that if w < B, t → f t (w) is right
continuous.
Let t be fixed. Let tn ↓ t. Then, if w < B, f tn (w) → f t (w).
By proposition (6, § 2, 86),
∀λ w, ∀s λws (B) = 0.
Hence, ∀λ w, ∀s, ∀λws w′ , f tn (w′ ) → f t (w′ ). Since by proposition (6, § 2,
87), ∀λ w, ∀ s, λws is a probability measure, and since f is bounded, it
follows by the dominated convergence theorem, that
∀λ w, ∀s, λws ( f tn ) → λws ( f t ).
Proposition 90. Let f be a bounded regular supermartingale adapted
to (C t )t∈R . Then, ∀λ w, ∀ s, t → λws ( f t ) is a decreasing function in
[s, +∞).
Proof. Let {t, t′ } be a fixed pair of real numbers, t < t′ . Since f is a
supermartingale adapted to (C t )t∈R , we have
′
∀ s, s ≤ t, ∀λ w, λws ( f t ) ≤ λws ( f t ).
6. Regular Disintegrations
100
′
Therefore, ∀λ w, ∀s ∈ Q s ≤ t, λws ( f t ) ≤ λws ( f t ).
99
′
Now, (w, s) → λws ( f t ) and (w, s) → λws ( f t ) are regular supermartin′
gales. Hence, ∀λ w, s → λws ( f t ) and s → λws ( f t ) are right continuous.
′
Therefore ∀λ w, ∀s < t, λws ( f t ) ≤ λws ( f t ).
Now, ∀λ w, λtw ( f t ) = f t (w). Since f is a supermartingale,
′
∀λ w, λtw ( f t ) ≤ f t (w).
′
Hence, ∀λ w, λtw ( f t ) ≤ λtw ( f t ). Therefore,
′
∀λ w, ∀s ≤ t, λws ( f t ) ≤ λws ( f t ).
′
Thus, for every pair {t, t′ }t < t′ , ∀λ w, ∀s ≤ t, λws ( f t ) ≤ λws ( f t ).
Hence, ∀λ w, ∀ pair {t, t′ }, t ∈ Q t′ ∈ Q t < t′ ,
′
∀s ≤ t, λws ( f t ) ≤ λws ( f t ).
By the previous proposition (6, § 2, 89), ∀λ w, ∀s, t → λws ( f t ) is right
continuous. Therefore, ∀λ w, ∀ pair {t, t′ }, t, t′ , ∀s ≤ t,
′
λws ( f t ) ≤ λws ( f t ).
Thus, ∀λ w, ∀s, t → λws ( f t ) is a decreasing function in [s, +∞).
Theorem 91. Let f be a regular supermartingale adapted to (C t )t∈R .
Then ∀λ w, ∀s, t → λws ( f t ) is decreasing and right continuous in [s, +∞)
and
∀λ w, ∀s, λws ( f s ) = f s (w).
Proof. If f is a bounded regular supermartingale, then from the previous
propositions, proposition (6, § 2, 89) and proposition (6, § 2, 90), we see
that
∀λ w, ∀s, t → λws ( f t )
100
is a decreasing right continuous function in [s, +∞). Hence, ∀λ w, ∀s,
t → λws ( f t ) is decreasing and lower semi-continuous in [s, +∞). (At s,
we mean only right lower semi-continuity).
If f is an arbitrary regular supermartingale, not necessarily bounded,
∀m ∈ N, inf( f, m) is a bounded regular supermartingale. Hence,
2. Properties of regular disintegrations
101
∀m ∈ N, ∀λ w, ∀s, t → λws [{inf( f, m)}t ] is decreasing and lower semicontinuous in [s, +∞).
Since ∀t, {inf( f, m)}t ↑ f t , we have
∀s, ∀t, ∀w, λws ( f t ) = lim λws [{inf( f, m)}t ].
m→∞
Therefore, ∀λ w, ∀s, t → λws ( f t ) is decreasing and lower semi-continuous
in [s, +∞).
If a function, in an interval is decreasing and lower semi-continuous,
if is right continuous there.
Hence, ∀λ w, ∀s, t → λws ( f t ) is decreasing and right continuous in
[s, +∞).
Let us now prove that
∀λ w, ∀s, λws ( f s ) = f s (w).
We have ∀s, ∀λ w, λws ( f s ) = f s (w). Hence,
∀λ w, ∀s ∈ Q, λws ( f s ) = f s (w).
Since ∀λ w, s → f s (w) is right continuous, to prove the theorem, it
is sufficient to prove that ∀λ w, s → λws ( f w ) is right continuous. Now,
[ " k k + 1!
· n .
R=
2n
2
k∈Z
n∈N
Define ∀ n ∈ N, w ∈ Ω
fns (w)
=
λws ( f
k+1
(2n )
"
!
k k+1
if s ∈ n , n ..
2
2
We claim that ∀n, fn is a regular supermartingale adapted to (C t )t∈R . 101
∀n, the regularity of fn is clear, since (λws )w∈Ω is a regular disintegration
s∈R
of λ. We have to only prove that ∀ n ∈ N, fn is a supermartingale
adapted to (C t )t∈R .
6. Regular Disintegrations
102
Clearly, fn is adapted to (C t )t∈R , ∀n ∈ N. We have to only prove that
∀ n, ∀ t, t′ ∈ R, t < t′ , ∀ A ∈ Ct ,
Z
Z
′
fnt (w)dλ(w) ≤
fnt (w)dλ(w).
A
A
"
!
k k+1
This is clear, if both t and belong to the same interval n , n .
2
2
for some t ∈ Z. Actually, in this case, we have even equality as (s, w) →
k+1
s
λw ( f 2n ) is a martingale.
′
It is sufficient to prove the above
! when t and
" t belong to
!
" inequality
k+1 k+2
k k+1
,
,
two consecutive intervals, any t ∈ n , n and t′ ∈
2
2
2n
2n
because for other values of t and t′ , t < t′ , the above inequality will then
follow from the transitivity of the conditional expectations.
Again to prove the inequality, it is sufficient to prove it when t =
k+1
k/2n for some k ∈ Z and t′ =
, because of the transitivity of the
2n
k+1
conditional expectations and because of the fact that (s, w) → λws ( f 2n )
is a martingale; i.e. we have to only prove that ∀ n ∈ N, ∀ k ∈ Z,
n
∀ A ∈ C k/2
t′
Z
A
k+1 k+2
k+1
Z
n
n
k/2
n
2
λw ( f 2n )dλ(w).
( f 2 )dλ(w) ≤
λw
A
But this follows immediately from the fact that f is a supermartingale. It is easy to check that
∀ s, ∀n,
s
∀λ w, fns (w) ≤ fn+1
(w).
Therefore, since ∀n ∈ N, fn is regular, we have
s
(w).
∀λ w, ∀n, ∀s, fns (w) ≤ fn+l
102
Hence ∀λ w, ∀s, lim fns (w) exists. But
n→∞
2. Properties of regular disintegrations
103
∀λ w, ∀s, t → λws ( f t )
is right continuous in [s, +∞). Hence
∀λ w, ∀s, lim fns (w) = λws ( f s ).
n→∞
Since ( fn )n∈N is an increasing sequence of regular supermartingales,
it follows from the ‘Upper enveloppe theorem’ that
∀λ w, s → λws ( f s )
is right continuous.
We now state and prove another important theorem, which we call
the “theorem of trajectories”. In the course of the proof of this theorem, we need the concepts relating to “Well-measurable processes” and
“C -analytic sets” where C is a σ-algebra on Ω. We state about well
measurable processes what we need. For the definition and properties
of C -analytic sets, we refer the reader to P.A. Meyer [1], Chapter, 3 and
section 1 “Compact pavings and Analytic sets”.
By a stochastic process X on Ω, with values in a topological spaces
E, we mean a collection (X t )t∈R of mappings on Ω with values in E such
that ∀t, X t ∈ Ôλ . A stochastic process can be considered as a mapping
X from Ω × R to E such that ∀t ∈ R, X t ∈ Ôλ . A stochastic process X is
said to be adapted to (C t )t∈R if ∀t ∈ R, X t ∈ C t .
Definition 92. A stochastic process X on Ω with values in R is said to
be well-measurable if it belongs to the σ-algebra on Ω × R, generated
by all sets A ⊂ Ω × R such that (w, t) → χA (w, t) is regulated and right 103
continuous and is adapted to (C t )t∈R .
See definition D14 in page 156 and remark (b) in page 157 in P.A.
Meyer [1].
We need the following important fact.
If X is a right continuous adapted process on Ω with values in R,
then X is well-measurable.
For a proof see remark (c) in page 157, in P.A. Meyer [1].
6. Regular Disintegrations
104
Theorem 93 (The theorem of trajectories). Let X be a right continuous
stochastic process on Ω, adapted to (C t )t∈R , with values in a topological space E which has a countable family of real valued continuous
functions separating its points (for example, a completely regular Suslin
space). Then,
∀λ w, ∀t ∈ R ∀λtw w′ , ∀s ≤ t, X s (w′ ) = X s (w).
i.e. ∀λ w, ∀t, λtw is carried by the set of all w′ whose trajectories coincide
with those of w upto the time t.
Proof. If BE is the Borel σ-algebra of E, it is easy to see that BE is
countably separating. Let s be any fixed real number. Let h be a function
on Ω with values in E such that h ∈ C t for all t ≥ s. Then proceeding as
in the proof of proposition (3, § 7, 54) and proposition (6, § 2, 88), we
can prove that
∀λ w, ∀ t ≥ s, ∀λtw w′ , h(w′ ) = h(w).
Now X s ∈ C t ∀t ≥ s. Hence,
∀s, ∀λ w, ∀t ≥ s, ∀λtw w′ , X s (w′ ) = X s (w).
104
Therefore, ∀λ w, ∀s ∈ Q, ∀t ≥ s, ∀λtw w′ , X s (w′ ) = X s (w). Hence, ∀λ w,
∀t, ∀λtw w′ , ∀s ∈ Qs ≤ t, X s (w′ ) = X s (w),
Now, ∀λ w, s → X s (w) is right continuous and hence ∀λ w, ∀t, ∀λtw w′ ,
s → X s (w′ ) is right continuous. Hence
∀λ w, ∀t, ∀λtw w′ , ∀s < t, X s (w′ ) = X s (w).
(1)
To prove the theorem, we have to only prove that ∀λ w, ∀t, ∀λtw w′ ,
X t (w′ ) = X t (w).
First let us consider the case when E = R.
Let us assume that X is a regular supermartingale adapted to (C t )t∈R .
Let M be any real number > 0. Let X M = inf(X, M). Then, X M is also a
regular supermartingale. Therefore, by theorem (6, § 2, 91),
t
t
∀λ w, ∀t, λtw (X M
) = XM
(w).
2. Properties of regular disintegrations
105
t ) = X t (w).
Let w be such that λtw is a probability measure and λtw (X M
M
t
t
t
′
If X (w) ≥ M, then X M (w) = M and X M (w ) ≤ M for all w′ ∈ Ω. Since
t (w), it follows therefore that ∀ t w′ , X t (w′ ) = M = X t (w)
λtw (Xwt ) = X M
λw
M
M
and hence,
∀λtw w′ , X t (w′ ) ≥ M.
Since ∀λ w, ∀t, λtw is a probability measure and since ∀λ w, ∀t,
t ) = X t (w), we see that
λtw (X M
M
∀ M ∈ R, M > 0, ∀λ w, ∀t, if X t (w) ≥ M, then
∀λtw w′ , X t (w′ ) ≥ M.
Hence, ∀λ w, ∀t, ∀M ∈ Q, M > 0, if X t (w) ≥ M, then ∀λtw w′ , X t (w′ ) ≥
M. Therefore ∀λ w, ∀t, ∀λtw w′ , X t (w′ ) ≥ X t (w). But by theorem (6, § 2,
91),
∀λ w, ∀t, λtw (X t ) = X t (w).
Hence, ∀λ w, ∀t, ∀λtw w′ , X t (w′ ) = X t (w).
This proves the theorem, when X is a regular supermartingale.
105
Since an adapted right continuous decreasing process with values
in R, is a regular supermartingale, the theorem holds when X is such a
process. Therefore also, when X is a bounded adapted right continuous
increasing process. By passing to the limit, the theorem therefore also
holds when X is an adapted increasing right continuous process.
Now, let X be any regulated right continuous process with values in
R.
Let ∀w ∈ Ω and t ∈ R,
σ(w, t) = X(w, t) − X(w, t−).
σ is again an adapted process with values in R. Let α > 0.
Since ∀λ w, Xw is a regulated function on R, ∀λ w, the number of t’s
in any relatively compact interval of R for which σ(w, t) ≥ α is finite.
α
Let n ∈ N and w ∈ Ω and t ∈ R. Define M n (w, t) as the number of
τ’s, τ ∈ (−n, t] for which σ(w, τ) ≥ α, if t > −n and zero otherwise.
α
Then ∀λ w, ∀n, Mn (w, t) is an integer for all t and it is easy to see
that ∀w, for which Xw is regulated, given any t ∈ R, ∃ δ > 0 such that
6. Regular Disintegrations
106
Mn (w, .) is constant in [t, t + δ). Hence ∀λ w, t → Mn (w, t) is trivially a
right continuous function. Moreover, ∀λ w, ∀n, Mn (w′ , ·) is an increasing
function of t.
α
Hence, ∀ n ∈ N, M n is an increasing right continuous process on
Ω × R.
Let us show that it is adapted to the family (Cˆλt )t∈R ; i.e. let us show
that
α
∀ t ∈ R, ∀n ∈ N, w → Mn (w, t) ∈ Cˆλt .
106
To understand the ideas involved in this prove, the reader is referred
to the proof of T52 in page 71 if P.A. Meyer [1], where similar ideas are
used.
Any right continuous or a left continuous process Z adapted to
t
(C )t∈R is progressively measurable with respect to (C t )t∈R and hence
∀t, the restriction of Z to Ω × (−n, t] belongs to C t ⊗ B(−n, t] where
B(−n, t] is the Borel σ-algebra of (−n, t]. (see D45 in page 68 and T 47
in page 70 of P.A. Meyer [1]). Note that (w, t) → X(w, t−) is a left continuous process and hence σ is progressively measurable with respect to
(C t )t∈R .
α
α
Since M tn is an integer valued function, to prove that ∀ ∈ N, M tn ∈
t
Cˆλ , it is sufficient to show that ∀ m ∈ N, w : Mnt (w) ≥ m is a C t analytic set.
Let A be the subset of Ω × Rm+1 consisting of points (w, −n, s1 , s2 ,
. . . , sm ) where w ∈ Ω, −n < s1 < s2 . . . < sm ≤ t and
σ(w, s1 ) ≥ α, σ(w, s2 ) ≥ α, . . . , σ(w, sm ) ≥ α.
This set A ∈ C t ⊗ B m+1 (−n, t] where B m+1 (−n, t] is the Borel σalgebra of (−n, t] × (−n, t] . . . × (−n, t].
|
{z
}
m+1 times
This is because of the fact that σ is progressively measurable with
respect to (C t )t∈R .
α
The projection of A on Ω is precisely the set {w : M tn (w) ≥ m}.
Hence {w : Mnt (w) ≥ m} is a C t -analytic set. Hence
α
∀ n, ∀ t, M tn ∈ Cˆλt .
2. Properties of regular disintegrations
Let ∀t, Ft be the σ-algebra
107
T ˆu
Cλ . Then (Ft )t∈R is an increasing right
u>t
α
continuous family of σ-algebras on Ω, contained in Ôλ and ∀n ∈ N, M n
α
is adapted to (Ft )t∈R . Since M n is an increasing right continuous process
adapted to (Ft )t∈R , the theorem is valid for this process.
α
α
Hence ∀ n ∈ N, ∀α > 0, ∀λ w, ∀t, ∀λtw w′ , ∀s ≤ t, M n (w′ , s) =
M n (w, s).
This shows that ∀n ∈ N, ∀α > 0, ∀λ w, ∀t, ∀λtw w′ , for all s ≤ t, the
number of jumps in (−n, s] of magnitude greater than or equal to α are
the same for Xw and Xw′ .
Since this is true for all rational α > 0 and the same thing analo- 107
gously for all rational α < 0, it follows that ∀ n ∈ N, ∀λ w, ∀t, ∀λtw w′ , for
all s ≤ t, the number of jumps in (−n, s] of any given magnitude are the
same for Xw and Xw′ .
Hence ∀ n ∈ N, ∀λ w, ∀t, ∀λtw w′ , for all s ≤ t, Xw′ and Xw have jumps
precisely at the same points in (−n, s] and the magnitudes of the jumps
at each point of a jump are the same for Xw′ and Xw .
Since this is true for every n ∈ N, we therefore have that
∀λ w, ∀t, ∀λtw w′ , Xw′
has a jump at a point s ≤ t if and only if Xw has one at s and the magnitudes of the jumps for Xw and Xw′ at s are the same.
Let C = {w ∈ Ω | ∀t, ∀λtw w′ , w′ has a jump at a point s ≤ t if and
only if Xw has one at s and the magnitudes of the jumps for Xw′ and Xw
at s are the same }. Then, by what we have seen, λ is carried by C.
Let B = {w ∈ Ω | ∀t, ∀λtw w′ , ∀s < t, X s (w′ ) = X s (w)}. Then, from
(1), λ is carried by B.
Let w ∈ B ∩ C. Let t be any point of R. Let Xw be continuous at t.
We have, since w ∈ B,
∀λtw w′ , ∀s < t, X s (w′ ) = X s (w).
Hence, ∀λtw w′ , lim X s (w′ ) exists and is equal to X t (w), since Xw is
s→t
s<t
continuous at t.
6. Regular Disintegrations
108
Let us prove that
∀λtw w′ , lim X s (w′ ) = X t (w′ ).
s→t
s<t
108
If not, the set Btw = {w′ ∈ Ω |
P
X s (w′ ) , X t (w′ )} is of positive
s→t
s<t
λtw -measure. But, since w ∈ C and since Xw is continuous at t and hence
has no jump at t, on no set of positive λtw -measure, all the w′ can have a
jump at t. Hence Btw must be of λtw -measure zero. Hence,
∀λtw w′ , lim X s (w′ ) = X t (w′ ) = X t (w).
s→t
s<t
Therefore, ∀λtw w′ , X t (w′ ) = X t (w).
Now, let w ∈ B ∩ C and t be a point at which Xw is discontinuous.
Then Xw has a jump at t and since Xw is right continuous at t, the jump
is equal to X(w, t) − X(w, t−). Hence, since w ∈ C, ∀λtw w′ , w′ has also a
jump at t and
X(w′ , t) − X(w′ , t−) = X(w, t) − X(w, t−).
But since w ∈ B,
∀λtw w′ , ∀s < t, X s (w′ ) = X s (w)
Hence
∀λtw w′ , X(w′ , t−) = X(w, t−).
Therefore,
∀λtw w′ , X(w′ , t) = X(w, t).
Since λ is carried by B ∩ C and since t ∈ R is arbitrary, we have
∀λ w, ∀t, ∀λtw w′ , X t (w′ ) = X t (w).
This combined with (1) gives that ∀λ w, ∀t, ∀λtw w′ , ∀s ≤ t, X s (w′ ) =
X s (w).
Hence the theorem is proved when X is an adapted, regulated, right
continuous real valued process.
2. Properties of regular disintegrations
109
109
Let C
= {A ⊂ Ω×R | χA is adapted and the theorem is true for the process χA }.
Then C is a σ-algebra on Ω × R.
C contains by the preceding, all the sets A for which χA is an adapted
regulated right continuous process. Hence C contains all the wellmeasurable processes.
Since a right continuous, adapted process is well-measurable, the
theorem is true for any right continuous adapted process with values in
R.
Now, if ( fn )n∈N is a countable family of continuous functions on E,
separating the points of E, and if X is a right continuous adapted process
with values in E, then ∀ n ∈ N, fn ◦ X is a real valued, adapted right
continuous process and hence the theorem is true for fn ◦ X, ∀n ∈ N.
Since the fn ’s separate the points of E and are countable, the theorem is
true for X.
Theorem 94. Let O be countably generated. Then,
Z
∀λ w, ∀s, ∀t, λtw′ λws (dw′ ) = λwMin(s,t) .
Proof. ∀B ∈ O, ∀s, t, s ≤ t,
Z
∀λ w,
λtw′ (B)λws (dw′ ) = λws (B)
since (t, w′ ) → λtw′ (B) is a martingale. Hence,
∀ B ∈ O, ∀t, ∀λ w, ∀s ∈ Q, s ≤ t,
Z
λtw′ (B)λws (dw′ ) = λws (B).
Since (λws )w∈Ω is a regular disintegration, ∀λ w, s → λws (B) and s →
s∈R
R
λtw′ (B)λws (dw′ ) are right continuous. Hence,
∀ B ∈ O, ∀t, ∀λ w, ∀s < t,
6. Regular Disintegrations
110
Z
λtw′ (B)λws (dw′ ) = λws (B).
Therefore,
110
∀B ∈ O, ∀λ w, ∀t ∈ Q, ∀s < t,
Z
λtw′ (B)λws (dw′ ) = λws (B).
Since (λtw )w∈Ω is a regular disintegration, ∀B ∈ O, ∀λ w, t → λtw (B) is
t∈R
right continuous. Hence ∀B ∈ O, ∀λ w, ∀s, ∀λws w′ , t → λtw′ (B) is right
continuous. Hence,
∀B ∈ O, ∀λ w, ∀t, ∀s < t,
Z
λtw′ (B)λws (dw′ ) = λws (B)
(1)
Since (λws (B))w∈Ω is an adapted right continuous process with values
s∈R
in R, by the previous theorem (6, § 2, 93),
∀ B ∈ O, ∀λ w, ∀s, ∀λws w′ , ∀t ≤ s, λtw′ (B) = λtw (B).
Therefore,
∀λ w, ∀s, ∀t ≤ s,
Z
λtw′ (B)λws (dw′ ) = λtw (B),
since ∀λ w, ∀s, λws is a probability measure.
From (I) and (II), we see therefore that
Z
∀B ∈ O, ∀λ w, ∀s, ∀t,
λtw′ (B)λws (dw′ ) = λwMin(s,t) (B).
II
Since O is countably generated, by the Monotone class theorem,
Z
∀λ w, ∀s, ∀t, ∀B ∈ O, λtw′ (B)λws (dw′ ) = λwMin(s,t) (B).
Hence,
∀λ w, ∀s, ∀t,
Z
.
λtw′ λws (dw′ ) = λMin(s,t)
w
Chapter 7
Strong σ-Algebras
1 A few propositions
We shall prove in this section two propositions which will be used later 111
on.
Proposition 95. Let (Ω, O, λ) be a measure space and C , a σ-algebra
contained in Ôλ . Let λ restricted to C be σ-finite. Then, the following
are equivalent:
(i) λ is disintegrated with respect to C by the constant measure valued function w → λCw = λ.
(ii) λ is ergodic on C i.e. ∀A ∈ C , λ(A) = 0 or 1.
Proof. Let us assume (i) and prove (ii).
By the definition of a measure space, λ . 0. (i) implies that λ is a
probability measure. By proposition (3, § 7, 49), for any disintegration
(λCw )w∈Ω of λ with respect to C , given A ∈ C , ∀λ w, λCw is carried by A or
by ∁A, according as w ∈ A or w ∈ ∁A. Therefore, given A ∈ C , either
λ(A) = 0 or if λ(A) > 0, there exists a w ∈ A such that λCw is carried
by A. Applying this to our special case where λCw = λ ∀w ∈ Ω, we
see that if λ(A) > 0, λ is carried by A and hence λ(A) = 1, since λ is a
probability measure. Thus, λ is ergodic on C .
111
7. Strong σ-Algebras
112
112
Let us now assume (ii) and prove (i). (ii) implies that λ is a probability measure.
To verify that λ is disintegrated with respect to C by the constant
measure valued function w → λCw = λ, according to proposition (3, § 7,
49), it is sufficient to verify that
(1) for any A ∈ O, w → λ(A) belongs to C .
R
(2) λ = λdλ(w)
(3) Given A ∈ C , λ is carried by A or by ∁A.
(1) and (2) are clear and (3) follows from the fact that λ is ergodic.
Proposition 96. Let (X, X, µ) be a measure space. Let x → ν x be a
measure valued function
on X with values in a measurable space (Ω, O),
R
ν ∈ X. Let ρ = ν x µ(dx). Let C be a σ-algebra contained in Ôρ ∩
T
Ôνx . Let ∀µ x, ν x have (λCw )w∈Ω as disintegration with respect to C .
x∈X
Then ρ also has (λCw )w∈Ω as disintegration with respect to C .
Proof. We have to only prove that
∀A ∈ C , χA · ρ =
Z
λCw dρ(w).
A
Let B ∈ O.
Z
χA · ρ(B) = ρ(A ∩ B) =
ν x (A ∩ B)dµ(x).
Z
∀µ x, ν x (A ∩ B) =
λCw (A ∩ B)dν x (w).
R
λCw (A ∩ B)dν x (w) = ν x (λC .(A ∩ B)) = ν x (χA · λC .(B)). Therefore,
Z
Z
ν x (A ∩ B)dµ(x) =
ν x (χA · λC .(B))µ(dx)
Z
=
χA (w) · λCw (B)ρ(dw)
2. Strong σ-Algebras
113
Z
= ( λCA ρ(dw))(B).
A
Since B ∈ O is arbitrary,
χA · ρ =
Z
λCw ρ(dw).
A
2 Strong σ-Algebras
Hereafter throughout this chapter, we shall make the following assump- 113
tions regarding Ω, O and λ.
(1) Ω is a topological space and O is its Borel σ-algebra
(2) O is countably generated
(3) λ is a probability measure on O and
(4) Ω has the λ-compacity metrizability property.
For example if Ω is a Suslin space and O is its Borel σ-algebra, (2)
is verified and (4) is also true for any probability measure λ.
Let Ō be the σ-algebra of all universally measurable sets of Ω. i.e.
T
Ō =
Ô where m+ (Ω) where is the set of all probability measures
µ∈m+ (Ω)
on Ω.
We remark that our assumptions regarding Ω, O and λ implies the
existence and uniqueness of disintegrations of λ with respect to any σalgebra contained in Ō.
If H is any σ-algebra contained in Ō, let (λH
w )w∈Ω denote a disintegration of λ with respect to H .
Definition 97. Let S and C be two σ-algebras contained in Ō. We say
C is λ-stronger than S if ∀λ w, λCw admits as disintegration with respect
to C , the family of measures (λCw′ )w′ ∈Ω .
The following proposition shows that the property “C is λ-stronger
than S ” does not depend on the chosen disintegrations.
7. Strong σ-Algebras
114
114
Proposition 98. Let S and C be two σ-algebras contained in Ō. Let
(λCw )w∈Ω (resp. (λCw )w∈Ω ) and (χCw )w∈Ω (resp. (χCw )w∈Ω ) be two disinteC
grations of λ with respect to S (resp. C ). If ∀λ w, λS
w has (λw′ )w′ ∈Ω
C
for disintegration with respect to C , then ∀λ w, χS
w has (χw′ )w′ ∈Ω for
disintegration with respect to C .
C
Proof. Since ∀λ w, λS
w has (λw′ )w′ ∈Ω for disintegration with respect to
C , we have
R
′
(1) ∀λ w, λCw = λCw′ dλS
w (dw )
w′ , λCw′ is carried by A or ∁A according
(2) ∀λ w, given A ∈ C , ∀λS
w
as w′ ∈ A or w′ ∈ ∁A.
To prove the proposition, we have only to prove that
R
′
χCw′ dχS
(i) ∀λ w, χS
w (w )
w =
w′ , χCw′ is carried by A or ∁A according as
(ii) ∀λ w, given A ∈ C , ∀χS
w
′
′
w ∈ A or w ∈ ∁A.
Because of the uniqueness of disintegrations, we have
S
(a) ∀λ w, λS
w = χw
(b) ∀λ w, λCw = χCw and therefore,
(c) ∀λ w, ∀χws w′ , λCw′ = χCw′ .
It is now easy to see that (i) and (ii) follow from (1) and (2) because
of (a) and (c).
Proposition 99. Let C be a σ-algebra contained in Ō. If C is countably
separating (in particular, if C is countably generated), it is λ-stronger
than all its sub σ-algebras.
Proof. Let S ⊂ C be a σ-algebra. Since O is countably generated, we
have
Z
′
∀λ w, λS
=
λCw′ λS
w (dw )
w
2. Strong σ-Algebras
115
115
because of the transitivity of the conditional expectations. Therefore,
since w′ → λCw′ belongs to C , to prove the proposition, it is sufficient
w′ , λCw′ is carried by
to prove by proposition (3, § 7, 52) that ∀λ w, ∀λC
w
the C -atom of w′ . Since (λCw )w∈Ω is a disintegration of λ with respect to
C and since C is countably separating, according to proposition (3, § 7,
w′ ,
53), ∀λ w′ , λCw′ is carried by the C -atom of w′ . Therefore, ∀λ w, ∀λS
w
λCw′ is carried by the C -atom of w′ .
Proposition 100. Let C and S be two σ-algebras contained in Ō. If C
is λ-stronger than S , it is λ-stronger than every sub σ-algebra of S .
Proof. Let S ′ be a σ-algebra contained
in S . Since O is countably
R
′
′
S
S
S
generated, we have ∀λ w, λw = λw′ dλw (w′ ) by the transitivity of the
conditional expectations.
C
′′
Since C is λ-stronger than S , ∀λ w′ , λS
w′ has′ (λw′′ )w ∈Ω for disinteS
′
S
gration with respect to C . Therefore, ∀λ w, ∀λw w , λw′ has (λCw′′ )w′′ ∈Ω
for disintegration with respect to C . Hence by proposition (7, § 1, 96),
′
C
∀λ w, λS
w has (λw′′ )w′′ ∈Ω for disintegration with respect to C .
This proves that C is λ-stronger than S ′ .
Definition 101. A σ-algebra C contained in Ō is said to be λ-strong if
it is λ-stronger than itself.
We see by the above proposition, proposition (7, § 2, 100), that a λstrong σ-algebra is λ-stronger than all its sub σ-algebras. From proposition (7, § 2, 99) we see that if a σ-contained in Ō is countably separating, it is µ-strong for any probability measure µ on O, if Ω has the
µ-compacity metrizability property.
Proposition 102. Let C be a σ-algebra contained in Ō. C is λ-strong 116
if and only if one of the two following equivalent conditions is true.
(i) ∀λ w, λCw is disintegrated with respect to C by the constant measure valued function w′ → λCw .
(ii) ∀λ w, λCw is ergodic on C .
7. Strong σ-Algebras
116
Proof. The equivalence of (i) and (ii) follows from proposition (7, § 1,
95).
Since O is countably generated, by Corollary (3, § 7, 56), we have
w′ , λCw′ = λCw .
∀λ w, ∀λC
w
Let C be λ-strong. Let us prove (i). ∀λ w, λCw is disintegrated by
w′ , λCw′ , λCw′ = λCw ,
(λCw′ )w′ ∈Ω with respect to C , and since ∀λ w, ∀λC
w
we see that ∀λ w, λCw is disintegrated by the constant measure valued
function w′ → λCw , with respect to C . This is (i)
Let us assume (i) and prove that C is λ-strong.
∀λ w, λCw is disintegrated with respect to C by the constant measure
w′ , λCw′ = λCw .
valued function w′ → λCw and ∀λ w, ∀λC
w
Hence, ∀λ w, λCw is disintegrated with respect to C by the family
C
(λw′ )w′ ∈Ω . This proves that C is λ-strong.
In the following proposition, we will be using the fact that if f is
a λ-integrable function and if (C n )n∈N is a decreasing sequence of σT n
n
algebras contained in Ō with C =
C , then ∀λ w, f C (w) → f C (w).
n=1
This follows immediately from the convergence theorem for martingales
with respect to a decreasing sequence of σ-algebras. See P.A. Meyer [1],
Chap. V. T21.
Proposition 103. Let (C n )n∈N be a decreasing sequence of σ-algebras
∞
T
contained in Ō and let C = C n . If ∀ n ∈ N, C n is λ-strong then C is
n=l
also λ-strong.
117
Proof. We have to prove that ∀λ w, λCw is disintegrated with respect to C
by (λCw′ )w′ ∈Ω . We have only to check that
Z
′
∀λ w, ∀B ∈ O, w →
χB (w′′ )λCw′ (dw′′ )
is a conditional expectation of χB with respect to C for the measure λCw .
i.e. we have to only prove that ∀λ w, ∀B ∈ O, for all A ∈ C ,
Z
Z Z
χB (w′ )λCw (dw′ ) = ( χB (w′′ )λCw′ (dw′′ ))λCw (dw′ ).
A
A
2. Strong σ-Algebras
117
To prove this, it is sufficient to prove that
Z
Z Z
′ C
′
∀ B ∈ O, ∀λ w,
χB (w )λw (dw ) = ( χB (w′′ )λCw′ (dw′′ ))λCw (dw′ )
A
A
for all A ∈ C . For if we prove this, an application of Monotone class
theorem will give the result, since O is countably generated.
Let B ∈ O. Since λ is a probability measure, χB is λ-integrable.
Hence,
n
∀λ w′ · (χB )C (w′ ) → (χB )C (w′ ).
R
R
n
i.e. ∀λ w′ , χB (w′′ )λCw′ (dw′′ ) → χB (w′′ )λCw′ (dw′′ ). Hence,
Z
Z
′′
′
′′ C n
χB (w′′ )λCw′ (dw′′ ).
w , χB (w )λw′ (dw ) →
∀λ w, ∀λC
w
Therefore, by Lebesgue’s dominated convergence theorem (which
n
w′ , λCw′ is a probability measure for all
is applicable here since ∀λ w, ∀λC
w
R
n
n and hence χB (w′′ )λCw′ (dw′′ ) ≤ 1 and 1 is integrable with respect to
λCw for λ-almost all w), we have ∀λ w, for all A ∈ C ,
Z Z
Z Z
′′
C
′
′′ C n
( χB (w )λw′ (dw ))λw (dw ) → ( χB (w′′ )λCw′ (dw′′ ))λCw (dw′ ).
A
A
Since ∀ n ∈ N, C n is λ-strong, and hence is λ-stronger than C , ∀λ w, 118
λCw is disintegrated with respect to C n by (λCw′ )w′ ∈Ω for all n ∈ N. Hence,
∀λ w, ∀B ∈ O, for all A ∈ C , for all n ∈ N,
Z
Z Z
n
χB (w′ )λCw (dw′ ).
( χB (w′′ )λCw′ (dw′′ ))λCw (dw′ ) =
A
A
Hence ∀B ∈ O, ∀λ w, for all A ∈ C ,
Z
Z Z
′ C
′
χB (w )λw (dw ) = ( χB (w′′ )λCw′ (dw′′ ))λCw (dw′ ).
A
This is what we wanted to prove.
7. Strong σ-Algebras
118
3 Choquet-type integral representations
In this section, we shall obtain a Choquet-type integral representation
for probability measures which have a given family of probability measures as disintegration with respect to a given σ-algebra of Ō.
Definition 104. Let C be a sub σ-algebra of Ō. C is said to be universally strong if it is λ-strong for every probability measure λ on O.
119
Let us fix a sub σ-algebra C of Ō which is universally strong,
throughout this section. Let (λw )w∈Ω be a family of probability measures on O. Let us assume that the measure valued function w → λw
belongs to C , i.e. ∀ B ∈ O, w → λw (B) ∈ C .
∀ w ∈ Ω, consider the set {w′ ∈ Ω | λw′ = λw }. This set ∈ C and
hence is a union of C -atoms. Hence, we shall call it the molecule of w
and write it as Mol w.
Let Ω̃ = {w ∈ Ω | λw is disintegrated with respect to C by (λw′ )w′ ∈Ω
and is carried by Mol w }.
Let K = {λ | λ a probability measure on O such that λ has (λw )w∈Ω
as disintegration with respect to C }.
Then K is a convex set.
R
Proposition 105. If λ ∈ K , λ is carried by Ω̃ and λ = λw dλ(w). If
R Ω
µ is any probability measure carried by Ω̃ and if ρ = λw dµ(w), then
Ω
ρ∈K.
Proof. Let λ ∈ K . Since C is universally strong, it is in particular
λ-strong λ has (λw )w∈Ω for disintegration with respect to C . Hence,
∀λ w, λw is disintegrated with respect to C by (λw′ )w′ ∈Ω . Moreover, by
Corollary (3, § 7, 56),
∀λ w, ∀λw w′ , λw′ = λw .
R
This shows that λ is carried by Ω̃. Since λ = λw λ(dw) and since λ
Ω
R
is carried by Ω̃, λ = λw λ(dw). The rest of the proposition follows
Ω̃
immediately from proposition (7, § 1, 96).
3. Choquet-type integral representations
119
Theorem 106. The extreme points of K are precisely the measures λw
where w ∈ Ω̃.
Proof. Let us first show that ∀w ∈ Ω̃, λw is extreme.
Let w ∈ Ω̃. Then λw ∈ K . Let µ and ν be two measures ∈ K and t
be a real number with 0 < t < 1 such that
λw = tµ + (1 − t)ν.
Mol w ∈ C and λw (Molw) = 1 since λw is carried by Mol w. Hence
µ and ν areR also carried by Mol Rw. By the previous proposition 7, § 3,
105), µ = λw′ µ(dw′ ) and ν = λw′ ν(dw′ ). Since µ and ν are carried
by Mol w, the integration is actually over only Mol w. Thus,
Z
Z
µ=
λw′ µ(dw′ ) and ν =
λw′ ν(dw′ ).
Mol w
Mol w
But for w′ ∈ Mol w, λw′ = λw and hence µ = λw = ν.
Thus λw is extreme ∀ w ∈ Ω̃.
120
Now, let λ be an extreme point of K . Let us show that there exists
a w ∈ Ω̃ such that λw = λ.
First of all, we claim that λ must be ergodic on C . For, if not,
∃A ∈ C such that 0 < λ(A) < 1,
λ = λ(A) ·
χ∁A · λ
χA · λ
+ λ(∁A) ·
.
λ(A)
λ(∁A)
χ∁A · λ
χA · λ
and
.
λ(A)
λ(∁A)
χ∁A · λ
χA · λ
Since A ∈ C , and λ ∈ K ,
also ∈ K . Similarly,
also
λ(A)
λ(∁A)
χA · λ
is
belongs to K . These two measures are not the same since
λ(A)
χ∁A · λ
carried by A and
is carried by ∁A. Thus, we get a contradiction
λ(∁A)
to the fact that λ is extreme. Hence λ must be ergodic on C . Hence
by proposition (7, § 1, 95) λ is disintegrated with respect to C by the
Thus, λ is a convex combination of the measures
7. Strong σ-Algebras
120
constant measure valued function w → λCw = λ. It follows therefore,
by uniqueness of disintegrations that ∀λ w, λw = λ. By the previous
proposition λ is carried by Ω̃. Hence there exists a w ∈ Ω̃ such that
λw = λ.
In view of theorem (7, § 3, 106) and proposition
(7, § 3, 105) we see
R
that the integral representation of λ ∈ K as λw λ(dw) is indeed of the
Ω̃
same type as the one considered by Choquet. But, we have not deduced
our result from Choquet’s theory.
Let us now prove a kind of uniqueness theorem.
Let Ω̃ be the quotient set of Ω̃ by the molecules. Let p be the canonical mapping from Ω̃ to Ω̃ . Let C ◦ be the σ-algebra on Ω̃ consisting of
sets whose inverse image under p belongs to C . If w ∈ Ω̃, let us denote
by ẇ the element p(w) of Ω̃ . ∀ẇ ∈ Ω̃ define the measure λẇ on C ◦
as the image measure of λw under the mapping p where w is such that
p(w) = ẇ.
This is independent of the choice of w in p−1 (ẇ), for if p(w1 ) =
p(w2 ), then λw1 = λw2 . If µ is any measure on C , let us denote by µ̇
its image measure under p on C ◦ . We note that if A ∈ C ◦ , λẇ (A) = 1
or 0 according as ẇ ∈ A or not. For, if ẇ ∈ A, then w ∈ p−1 (A) where
w is such that p(w) = ẇ. Hence Mol w ⊂ p−1 (A). Therefore, λẇ (A) =
λw (p−1 (A)) = 1 since λw is carried by Mol w. If ẇ < A, ẇ ∈ ∁A and by
the same argument λẇ (∁A) = 1. Hence λẇ (A) = 0.
•
•
•
•
121
•
Theorem 107. Let
R λ ∈ K . If µ is any probability measure carried by
Ω̃ such that λ = λw µ(dw) , then µ̇ = λ̇ on C ◦ .
R
R
Proof. Let λ = λw µ(dw). We first see that λ̇ = λẇ µ̇(dẇ). For let
A∈
C◦
Ω̃ •
Ω̃
λ̇(A) = λ(p−1 (A)) =
=
=
Z
Z
Z
λw (p−1 (A))µ(dw)
λẇ (A)µ(dw)
λ p(w) (A)µ(dw)
4. The Usual σ-Algebras
121
=
Hence,
λ̇ =
Z
Z
λẇ (A)µ̇(dẇ).
λẇ µ̇(dẇ).
Ω̃ •
For any A ∈ C ◦ , λ̇(A) =
R
λẇ (A)µ̇(dẇ). Since λẇ (A) = 1 or 0
Ω̃ •
according as ẇ ∈ A or ∁A, the integration is actually only over A; i.e.
Z
λ̇(A) =
λẇ (A)µ̇(dẇ).
A
But,
R
λẇ (A)µ̇(dẇ) = µ̇(A) since λẇ (A) = 1 for ẇ ∈ A. Hence,
A
λ̇(A) = µ̇(A), ∀A ∈ C ◦
and therefore, λ̇ = µ̇.
4 The Usual σ-Algebras
In this section, we shall define the σ-algebras that occur in the theory of 122
Brownian motion on the real line and prove some properties of them.
Let Ω[0,+∞) be the set of all real valued continuous functions on
[0, +∞). Let D be a countable dense subset of [0, +∞). For t ∈ [0, +∞)
define the mapping πt from Ω[0,+∞) to R as πt (w) = w(t), where w is an
element of Ω[0,+∞) . πt is called the t′ th projection. Let ID be the topology on Ω[0,+∞) which is the coarsest making all the (πt )t∈D continuous.
Let OD be the Borel σ-algebra of Ω[0,+∞) for the topology ID . It can be
easily checked that OD is the smallest σ-algebra making the projections
(πt )t∈D measurable. Since ∀t ∈ [0, +∞), πt = lim πtn , it follows immetn →t
tn ∈D
diately that OD is also the smallest σ-algebra making all the projections
(πt )t∈[0,+∞) measurable.
Let IP be the topology of pointwise convergence on Ω[0,+∞) i.e. IP
is the coarsest topology making all the (πt )t∈[0,+∞) continuous. Let OP be
122
123
7. Strong σ-Algebras
the Borel σ-algebra of Ω[0,+∞ in this topology IP . Let I be the topology
of uniform convergence on compact sets of [0, +∞). Let O be the Borel
σ-algebra of Ω[0,+∞) in this topology I. We easily see that ID is coarser
than IP which in turn is coarser than I . Hence OD ⊂ OP ⊂ O.
Now, Ω[0,+∞) is a separable Fréchet space under the topology I and
hence is a Polish space i.e. is homeomorphic to a complete separable
metric space. It is a theorem that the Borel σ-algebra of a Polish space
is the same as the Borel σ-algebra of any coarser Hausdorff topology.
Hence OD = OP = O.
Thus the smallest σ-algebra making all the (πt )t∈[0,+∞) measurable
coincides with the Borel σ-algebra of the topology of pointwise convergence on [0, +∞) and it is countably generated since O is countably
generated.
Let U ′ be the σ-algebra on Ω[0,+∞) generated by (π s )s≤t where t ∈
[0, +∞). Let Ω[0,t] be the space of all real valued continuous functions on
[0, t]. For 0 ≤ s ≤ t, let π′s be the map defined on Ω[0,t] as π′s (w) = w(s)
where w ∈ Ω[0,t] . As above, we can see that the Borel σ-algebra of Ω[0,t]
for the topology of pointwise convergence on [0, t] is the same as the
σ-algebra generated by (π′ s)0≤s≤t . Let us denote this σ-algebra by O t .
As above we can see that O t is countably generated.
Let p : Ω[0,+∞) → Ω[0,t] be the restriction map. It is easily checked
that U t is equal to p−1 (O t ) where p−1 (O t ) is the σ-algebra consisting of
sets p−1 (A) as A varies over O t . Since O t is countably generated, U t is
also countably generated.
Let w ∈ Ω[0,+∞) . The U t -atom of w is p−1 (p(w)) i.e. the set of all
trajectories which coincide with w upto time t.
Proposition 108. A ∈ U t ⇐⇒ A ∈ O and is a union of U t -atoms.
Proof. Let A ∈ U t . Then clearly A ∈ O and is a union of U t -atoms. We
have to prove only the other way.
Let ( fn )n∈N be a countable number of functions generating the σalgebra U t . Consider the mapping ϕ : Ω[0,+∞) → RN given by ϕ(w) =
( fn (w))n∈N . Since ∀ n ∈ N, fn ∈ U t , ϕ also ∈ U t and hence is Borel
measurable i.e. measurable with respect to O. Hence ϕ(Ω[0,+∞) ) is a
Suslin subset of RN since Ω[0,+∞) is a Polish space for the topology I
4. The Usual σ-Algebras
123
and O is its Borel σ-algebra. Since A ∈ O, A is Suslin and hence ϕ(A)
ia Suslin. Similarly ϕ(∁A) is also Suslin. Since A is union of U t -atoms,
it is easily checked that ϕ(Ω[0,+∞) )ϕ(A) is precisely ϕ(∁A). Thus the
Suslin subsets ϕ(A) and ϕ(∁A) are complementary in ϕ(Ω[0,+∞) ) and
hence are Borel. Since A is a union of U t -atoms, A = ϕ−1 (ϕ(A)). Since 124
ϕ ∈ U ′ and ϕ(A) is Borel, it follows that ϕ−1 (ϕ(A)) i.e. A ∈ U t .
T
Let ∀t ∈ [0, +∞), C t be the σ-algebra
U t+ǫ .
ǫ>0
Corollary 109. A ∈ C t =⇒ A ∈ O and is a union of C t -atoms.
Proof. If A ∈ C t , then A ∈ O and is a union of C t -atoms. Conversely if
A ∈ O and is a union of C t -atoms, it is ∀ǫ > 0, a union of U t+ǫ -atoms
and hence by the above proposition, A ∈ U t+ǫ ∀ǫ > 0. Hence AǫC t . Proposition 110. The C t -atom of w consists of precisely the trajectories
which coincide with w a little beyond t. i.e.
C t − atom of w = {w′ | ∃ tw′ > t such that w′ = w in [0, tw′ ]}.
Proof. Let B = {w′ | ∃ tw′ > t such that w′ = w in [0, tw′ ]}
[
B=
{w′ | w′ = w in [0, t + ǫ]}
ǫ>0
[
=
{U t+ǫ − atom of w}.
ǫ>0
Let A be any set ǫC t containing w. Then, ∀ǫ > 0, AǫU t+ǫ and hence
A contains the U t+ǫ -atom of w, ∀ǫ > 0. Hence A ⊃ B.
Therefore, to prove the proposition, sufficient to prove that B ∈ C t .
[
B=
{U t+ǫ − atom of w}
ǫ>0
=
[
1
{U n − atom of w}
t+
n∈N
=
[ [
m∈N 1 < 1
n m
n∈N
1
{U t+ n − atom of w}
7. Strong σ-Algebras
124
∀ m ∈ N,
S
1
1
{U t+ n − atom of w} ∈ U t+ m and hence,
1 1
n<m
n∈N
\ [
1
{U t+ n − atom of w}ǫC t .
m∈N 1 < 1
n m
n∈N
125
One can prove that ∀t, C t is not countably separating. For a proof
see L. Schwartz [1], page 161.
However, ∀t, C t is universally strong. For, since ∀ǫ > 0, U t+ǫ is
countably generated, it is universally strong by proposition (7, § 2, 99).
T t+ 1
U n , by proposition (7, § 2, 103), it follows that C t is
Since C t =
n∈N
universally strong ∀t ∈ [0, +∞).
5 Further results on regular disintegrations
In this section, let us assume that Ω is a compact metrizable space and
O is its Borel σ-algebra. Let (C t )t∈R be a right continuous increasing
family of σ-algebras contained in O which is the σ-algebra of all universally measurable sets of Ω. Let λ be a probability measure on O.
Note that a unique regular disintegration of λ with respect to (C t )t∈R
exits.
Let us assume that ∀t ∈ R, C t is λ-strong.
Theorem 111. Let (λtw )w∈Ω be a regular disintegration of λ with respect
t∈R
to (C t )t∈R . Then ∀λ w, ∀s, λws has a regular disintegration with respect
to (C t )t∈R given by
(t, w′ ) → λtw′ for t ≥ s
(t, w′ ) → λws for t < s.
Proof. Let t be fixed. Since C t is λ-strong, ∀λ w, λtw has (λtw )w′ ∈Ω for
disintegration with respect to C t . Since a set of λ-measure zero is also
∀λ w, ∀s, of λws -measure zero, it follows therefore that
5. Further results on regular disintegrations
C t.
125
∀λ w, ∀s, ∀λws w′ , λtw′ has (λtw′′ )w′′ ǫΩ for disintegration with respect to
By theorem (6, § 2, 94), we have
∀λ w, ∀s ≤ t,
λws
=
126
Z
λtw′ λws (dw′ ).
Hence, by proposition (7, § 1, 96), ∀λ w, ∀s ≤ t, λws has (λtw′′ )w′′ ∈Ω
for disintegration with respect to C t . Thus, ∀t, ∀λ w, ∀s ≤ t, λws has
(λtw′′ )w′′ ∈Ω for disintegration with respect to C t .
Therefore, ∀λ w, ∀t, a dyadic rational ∀ s ≤ t, λws has (λws ′′ )w′′ ∈Ω for
disintegration with respect to C t .
Now, let µ be a probability measure on O and let (δtw )w∈Ω be a disintegration of µ with respect to C t ∀ dyadic t.
For any t ∈ R, define

τ (t)


vague lim δwn , if it exists
t
δw = 

0, i f not
Then, since Ω is compact metrizable, we can easily check that
(δtw )w∈Ω is a regular disintegration of µ with respect to (C t )t∈R .
t∈R
Now we have,
∀λ w, ∀t ∈ R, λtw = vague lim λτwn (t) .
n→∞
Hence, since ∀λ w, ∀s, λws has a disintegration (λtw′′ )w′′ ∈Ω with respect
to C t for all t dyadic ≥ s, by the preceding paragraph, ∀−λw, ∀s, λws has
(λtw′′ )w′′ ∈Ω for a regular disintegration with respect to C t for all t ≥ s.
By the theorem of trajectories, i.e. by theorem (6, § 2, 93),
∀λ w, ∀s, ∀λws w′ , ∀t ≤ s, λtw′ = λtw .
In particular,
∀λ w, ∀s, ∀λws w′ , λws ′ = λws .
Since C s is λ-strong. ∀λ w λws is disintegrated by (λws ′ )w′ ∈Ω with 127
respect to C s . Since ∀λ w, ∀s, ∀λws w′ , λws ′ = λws , ∀λ w, ∀λ w, ∀s, λws is
7. Strong σ-Algebras
126
disintegrated with respect to C s by the constant measure valued function
w′ → λws . Hence, by proposition (7, § 1, 95), ∀λ w, ∀s, λws is ergodic on
C s . Hence ∀λ w, ∀s, ∀t ≤ s, λws is ergodic on C . Hence, again by the
proposition (7, § 1, 95), ∀λ w, ∀s, ∀t ≤ s, λws is disintegrated with respect
to C t by the constant measure valued function w′ → λws .
Thus, we have proved that ∀λ w, ∀s, λws has with respect to (C t )t∈R
a disintegration given by (t, w′ ) → λtw′ for t ≥ s and (t, w′ ) → λws for
t < s.
We have to check only that this is a regular disintegration. Since
(t, w) → λtw is a regular disintegration of λ, we have to check only the
right continuity at the point s. i.e., we have to only prove that
∀ B ∈ O, ∀λ w, ∀s, ∀λws w′ , lim λtw′ (B) = λws (B).
t↓s
But this follows immediately from the fact that ∀λ w′ , lim λtw′ (B) =
t↓s
λws ′ (B) and
∀λ w, ∀s, ∀λws w′ , λws ′ = λws .
Corollary 112. Let f be a regular supermartingale adapted to (C t )t∈R .
Then ∀λ w, ∀s, f remains a supermartingale on the set Ω × [s, +∞), for
the measure λws .
Proof. From the above theorem, we have ∀λ w, ∀s, ∀ pair {t, t′ } with
s ≤ t ≤ t′ , ∀λws w′ .
s
′
′
E λw ( f t | C t )(w′ ) = λtw′ ( f t )
s
128
′
where E λw ( f t | C t )(w′ ) stands for the value at w′ of the conditional
′
expectation of f t with respect to C t for the measure λws .
′
Because f is a supermartingale, ∀λ w, λtw ( f t ) ≤ f t (w). Hence
′
∀λ w, ∀s, ∀λws w′ · λtw′ ( f t ) ≤ f t (w′ ).
s
′
Hence ∀λ w, ∀s, ∀ pair {t, t′ } with s ≤ t ≤ t′ , ∀λws w′ , E λw ( f t |
t
C )(w′ ) ≤ f t (w′ ).
5. Further results on regular disintegrations
127
This shows that ∀λ w, ∀s, f is a supermartingale on Ω × [s, +∞) for
the measure λws .
The regularity of f with respect to λws follows from that of f with
respect to λ.
Remark 113. Under the assumptions that C t is λ-strong ∀ t ∈ R and
Ω, compact metrizable, note that the above corollary is stronger than
theorem (6, § 2, 91).
Bibliography
[1] Blumenthal, R. M. and Getoor, R.K. : Markov Processes and Po- 129
tential theory. (Acad. Press, 1968).
[2] Dieudonné, J. : Sur le théorème de Lebesgue Nikodym III. (Ann.
Univ. Grenoble Sec. Sci. Math. PHys. (N-S) 23, (1948), 25-53).
[3] Doob, J. L. : Stochastic Processes. (John Wiley & Sons 1967).
[4] Hille, E, and PHillips, R.S. : Functional Analysis and SemiGroups. (Amer. Math. Soc. Colloq. Publications Vol. XXXI,
1957).
[5] Jirina, M. : On Regular Conditional Probabilities. (Czech. Math.
Journal Vol. 9, (1959), 445-450).
[6] Maharam, D. : On a theorem of Von Neumann. (Proc. Amer. Math.
Soc. Vol 9(1958), 987-994).
[7] Meyer, P.A. : Probability and Potentials. (Blaisdell publishing
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[8] Schwartz, L. : Surmartingales réglières à valeurs mesures et
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129

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