Dedicated to the memory of Helmut H. Schaefer
VECTOR INTEGRALS AND A POINTWISE
MEAN ERGODIC THEOREM
FOR TRANSITION FUNCTIONS
RADU ZAHAROPOL
Our goal is to discuss the use of a certain vector integral in order to extend the
weak* and pointwise mean ergodic theorems for equicontinuous Markov-Feller
pairs to equicontinuous Feller transition functions.
AMS 2000 Subject Classification: Primary 28B05; Secondary 26A42, 60J35.
Key words: Bochner integral, Markov operator, mean ergodic theorem, pointwise
convergence, transition function, weak* convergence.
PROLOGUE
The main ideas of the paper can be told in the language of fairy tales: We
consider Cinderella (in Romanian we call her “Cenuşăreasa”) and a handsome
prince (in Romanian, “Făt-Frumos”). Cinderella (the pointwise integral that
will be defined in the paper) is a modest girl without education (knows almost
no measure theory, let alone ergodic theory), but she is easygoing and exists
under almost any conditions. By contrast, the prince (the Bochner integral)
has a very solid measure theoretical foundation and (thanks to Dunford and
Schwartz [3]) is familiar with ergodic theory; however, the prince is spoiled,
and exists under rather sophisticated conditions. Together, Cinderella and the
prince can help us deal with two ergodic theorems. And then, of course, they
will live happily ever after (again, in Romanian we say “Am ı̂ncălecat pe-o şea
şi v-am spus povestea aşa.”).
1. INTRODUCTION
In [6] and [8] we have obtained various results concerning transition probabilities: an extension of an ergodic decomposition that goes back to Kryloff,
Bogoliouboff, Beboutoff and Yosida, results on supports of various types of
invariant probabilities, a weak* mean ergodic theorem, and a pointwise mean
REV. ROUMAINE MATH. PURES APPL., 53 (2008), 1, 63–78
64
Radu Zaharopol
2
ergodic theorem. In recent years we have extended all the above-mentioned
results to transition functions, and currently we are writing the final form of a
small monograph on transition functions that will include these extensions. As
expected, the proofs of the extensions are more sophisticated than the proofs
of the corresponding results for transition probabilities as one encounters various difficulties due to the transition function setting. Our goal in this note
is to show how to overcome one such difficulty when extending the pointwise
and the weak* mean ergodic theorems. In doing so, we will discuss a vector
integral; it seems that the integral has never been mentioned explicitly in the
literature, but, in all likelihood, it has been used implicitly earlier.
Let (X, d) be a locally compact separable metric space and let B(X) be
the σ-algebra of all Borel subsets of X.
As usual, we say that a map P : X ×B(X) → R is a transition probability
if the following two conditions are satisfied.
(i) For every x ∈ X, the map µx : B(X) → R defined by µx (A) = P (x, A)
for every A ∈ B(X) is a probability measure.
(ii) For every A ∈ B(X), the function gA : X → R defined by gA (x) =
P (x, A) for every x ∈ X is Borel measurable.
Let Bb (X) be the Banach lattice (under the sup (uniform) norm and
the pointwise order) of all real-valued bounded Borel measurable functions
defined on X, and let M(X) be the Banach lattice of all real-valued signed
Borel measures on X, the norm on M(X) being the total variation norm, and
the order on M(X) being the usual one (for the definition and a comprehensive
treatment of Banach lattices, see the classical monograph of Schaefer [5]).
Given a transition probability P on (X, d), we can define two operators
S : Bb (X) → Bb (X) and T : M(X) → M(X) by
Z
(1.1)
Sf (x) =
f (y)P (x, dy)
X
for every f ∈ Bb (X) and x ∈ X, where P (x, dy) stands for dµx (y), µx being
the probability measure defined by P and x ∈ X that appears at (i) in the
definition of a transition probability, and by
Z
(1.2)
T µ(A) =
P (x, A) dµ(x)
X
for every µ ∈ M(X) and A ∈ B(X).
It is easy to see that S and T are well-defined in the sense that the
function Sf obtained using (1.1) belongs to Bb (X) whenever f ∈ Bb (X), and
that if T µ is given by (1.2), then T µ belongs to M(X) whenever µ ∈ M(X).
Also easy to see is the fact that S and T are linear positive contractions
3
Vector integrals and a pointwise mean ergodic theorem
65
of Bb (X) and M(X), respectively. We call the pair (S, T ) the Markov pair
defined by P . For details on Markov pairs, see [8].
Let Cb (X) be the Banach sublattice of Bb (X) of all real-valued continuous bounded functions defined on X.
A pair (S, T ) is called a Markov-Feller pair if Sf belongs to Cb (X) whenever f ∈ Cb (X). If (S, T ) is a Markov-Feller pair, the corresponding transition
probability P is called a Feller transition probability. For details on MarkovFeller pairs, see [6] and [8].
A family (Pt )t∈[0,+∞) of transition probabilities defined on (X, d) is called
a transition function if it has the property that
Z
(1.3)
Ps+t (x, A) =
Ps (y, A)Pt (x, dy)
X
for every s ∈ [0, +∞), t ∈ [0, +∞), A ∈ B(X) and x ∈ X. Note that (1.3) is
the familiar Chapman-Kolmogorov equation.
Let (Pt )t∈[0,+∞) be a transition function. For each t ∈ [0, +∞) we
can construct the Markov pair (St , Tt ) defined by Pt . We call the family
((St , Tt ))t∈[0,+∞) constructed in this way the family of Markov pairs defined
(or generated) by (Pt )t∈[0,+∞) . It can be shown that the families (St )t∈[0,+∞)
and (Tt )t∈[0,+∞) are semigroups of operators (on Bb (X) and M(X), respectively).
We say that (Pt )t∈[0,+∞) is a Feller transition function if (St , Tt ) is a
Markov-Feller pair for every t ∈ [0, +∞). In this case, we also say that
(St )t∈[0,+∞) is a Feller semigroup of operators.
We say that (Pt )t∈[0,+∞) satisfies the standard measurability assumption
(s.m.a.) if for every A ∈ B(X), the map (t, s) 7→ Pt (x, A), (t, x) ∈ [0, +∞)×X,
is jointly measurable with respect to t and x (that is, the map is measurable
with respect to the product σ-algebra L([0, +∞)) ⊗ B(X), where L([0, +∞))
is the σ-algebra of all Lebesgue measurable subsets of [0, +∞)).
Let C0 (X) be the Banach sublattice of Cb (X) (and, of course, of Bb (X))
of all real-valued continuous functions on X that vanish at infinity.
It can be shown that a Feller transition function which is C0 (X)-strongly
continuous satisfies the s.m.a.
We say that (Pt )t∈[0,+∞) (or (St )t∈[0,+∞) , or ((St , Tt ))t∈[0,+∞) ) is C0 (X)strongly continuous if for every f ∈ C0 (X), the map t 7→ St f , t ∈ [0, +∞),
is continuous with respect to the topology of uniform convergence on Bb (X)
and the usual standard topology on [0, +∞).
We say that (Pt )t∈[0,+∞) (or (St )t∈[0,+∞) , or ((St , Tt ))t∈[0,+∞) ) is C0 (X)equicontinuous if for every f ∈ C0 (X), for every convergent sequence (xn )n∈N
of elements of X, and for every ε ∈ R, ε > 0, there exists nε ∈ N such that
|St f (xn ) − St f (x)| < ε for every t ∈ [0, ∞) and for every n ≥ nε , where
x = limn→∞ xn .
66
Radu Zaharopol
4
Now, assume that (Pt )t∈[0,∞) satisfies the s.m.a. It can be shown that
for every f ∈ Bb (X) and every x ∈ X, the map t 7→ St fR(x), t ∈ [0, ∞) is meas
surable. Since the map is also bounded, the integral 0 St f (x) dt exists and
is a real number for every s ∈ [0, ∞). Hence, it makes sense to try to single
out a class of transition functions (Pt )t∈[0,∞) that satisfy the s.m.a. and have
Rs
the property that lim 1s 0 St f (x) dt exists for every f ∈ C0 (X) and x ∈ X,
s→∞
thus extending the pointwise ergodic theorem for Feller transition probabilities that we proved in [6], Section 4.3, Corollary 4.3.2. However, at least for
the time being, the only manner in which we can obtain such an extension is
to obtain first an extension of the weak* mean ergodic theorem for transition
probabilities discussed in Theorem 4.3.1 of [6] to transition functions.
In order
Rs
to obtain such an extension, we have to assign a meaning to 0 Tt µ dt for every
µ ∈ M(X) and s ∈ [0, +∞) in order
to be able to obtain conditions that inR
1 s
sure the existence of w*- lim s 0 Tt µ dt for every µ ∈ M(X), where w*-lim
s→+∞
stands forRthe limit in the weak* topology of M(X). Ideally, we would like the
s
integrals 0 Tt µ dt, s ≥ 0, to be Bochner integrals (for details on the Bochner
integral, see Dinculeanu’s comprehensive monograph [2] on vector and stochastic integrals and Appendix E of the excellent textbook in measure theory
of Cohn [1]) because in such a case we could try to use results from the first
volume of Dunford andR Schwartz’s monograph [3] to study conditions for the
s
existence of w*- lim 1s 0 Tt µ dt, µ ∈ M(X). However, in general (actually,
s→∞
in most cases of interest), given µ ∈ M(X), the map t 7→ Tt µ, t ≥ 0, is not
strongly measurable because the range of the map fails to be separable (for
the terminology used here, see Appendix E of Cohn [1]). Fortunately, we can
consider a very simple vector integral that we call the pointwise (vector) integral, and that can be used to state and prove a weak* mean ergodic theorem
which in turn allows us to prove the following result.
Theorem 1.1 (Pointwise Mean Ergodic Theorem for Transition Functions). Let (Pt )t∈[0,+∞) be a Feller transition function that is C0 (X)-strongly
continuous and C0 (X)-equicontinuous (note that, as pointed out earlier,
such
Rs
a transition function satisfies the s.m.a., as well ). Then lim 1s 0 St f (x) dt
s→∞
exists and is a real number whenever f ∈ C0 (X) and x ∈ X.
Remark. It is easy to find transition functions that satisfy the conditions
of the above theorem, and, therefore, for which the conclusion of the theorem
holds true. A particularly large and interesting family of examples can be
obtained as follows (for details on unexplained terminology, see [7]).
Assume that (X, d) is a locally compact separable metric semigroup with
identity u. We use ∗ to denote the convolution operation on M(X). If µ ∈
M(X), then we use the notation µ0 = δu (δu is the Dirac measure concentrated
5
Vector integrals and a pointwise mean ergodic theorem
67
at u; throughout the paper, δx is the Dirac measure concentrated at x ∈ X),
µ1 = µ, and µn = µ ∗ µ ∗ · · · ∗ µ for n ∈ N, n ≥ 2. Let µ ∈ M(X) be
|
{z
}
n times
a probability measure. It can be shown that the series
∞
P
k=0
αk k
k! µ
converges
in the norm topology of M(X) to a positive element να of M(X) and that
∞
P
1
kνα k = eα (where, of course, e =
k! ) whenever α ∈ R, α > 0. Set µ0 = δu ,
and µα =
e−α
∞
P
k=0
k=0
αk k
k! µ
for every α ∈ R, α > 0. Then µα , α ∈ [0, +∞) are
probability measures, and it can be shown that µα+β = µα ∗ µβ for every
α ∈ [0, +∞) and β ∈ [0, +∞) (the family (µα )α∈[0,+∞) is an example of a
one-parameter convolution semigroup; for a comprehensive discussion of such
semigroups when X is a group, see Chapter III and Chapter IV of Heyer
[4], a standard reference to many topics including one-parameter semigroups).
For every α ∈ [0, +∞), let Pα be the transition probability defined by µα
(see [7] for details). It can be shown that (Pα )α∈[0,+∞) is a Feller transition
function. If µ is an equicontinuous probability measure, then it can be shown
that (Pα )α∈[0,+∞) satisfies all the conditions of Theorem 1.1. Thus, if µ is
equicontinuous, the conclusion of the theorem holds true for (Pα )α∈[0,+∞) .
In particular, the conclusion of Theorem 1.1 holds true for any transition
function (Pα )α∈[0,+∞) defined by any probability measure µ whenever (X, d)
is a compact metric semigroup. Our goal in the next section is to define the above-mentioned vector
integral and to discuss several of its properties that are useful for proving
Theorem 1.1. In order to keep the length of this paper within reasonable limits,
we will not prove Theorem 1.1 here. The proof of Theorem 1.1 will be included
in the monograph that we mentioned earlier that we are currently writing.
2. THE POINTWISE VECTOR INTEGRAL
Let (Ω, Σ, ν) be a measure space, let Y be a nonempty set, and let F be a
collection of real-valued functions defined on Y (usually F is a Banach lattice
of such functions). We say that a function ϕ : Ω → F is pointwise measurable
if the condition below is satisfied.
(PM) For every y ∈ Y , the function ϕy : Ω → R defined by ϕy (t) =
ϕ(t)(y) for every t ∈ Ω is Σ-measurable.
A pointwise measurable function ϕ : Ω → F is said to be pointwise
integrable if the two conditions below are satisfied.
(PI1) For every y ∈ Y , the function ϕy , defined at (PM), is integrable.
68
Radu Zaharopol
6
R
(PI2) The function I : Y → R defined by I(y) = Ω ϕ(t)(y) dν(t) for
every y ∈ Y belongs to F.
If ϕ is pointwise integrable, then the function I defined
at (PI2) is
R
called the pointwise integral of ϕ andR is denoted by P- Ω ϕ(t) dν(t). For
y ∈R Y , we will use the notation P- Ω ϕ(t)(y) dν(t) for I(y), rather than
P- Ω ϕ(t) dν(t) (y), which is the formally correct notation, but it is cumbersome.
Remark A. We mentioned earlier that we believe strongly that the pointwise integral has appeared often in implicit form in the literature earlier. In
support of the above assertion, let us discuss briefly how one can state a restricted form of Fubini’s theorem in terms of pointwise integrals. Let (Y, Y, µ1 )
and (Z, Z, µ2 ) be two finite measure spaces, let Y ⊗Z be the product σ-algebra
of Y and Z on Y × Z, and let µ1 ⊗ µ2 be the product measure of µ1 and µ2 on
the measurable space (Y × Z, Y ⊗ Z). Now, let g : Y × Z → R be a bounded
Y ⊗ Z-measurable function. For every y ∈ Y , let gy : Z → R be defined by
gy (z) = g(y, z) for every z ∈ Z and, for every z ∈ Z, let g(z) : Y → R be
defined by g(z) (y) = g(y, z) for every y ∈ Y . Finally, let Bb (Y ), Bb (Z), and
Bb (Y × Z) be the Banach spaces of all real-valued bounded measurable functions defined on Y , Z, and Y × Z, respectively. Now let ϕ : Y → Bb (Z)
and ψ : Z → Bb (Y ) be defined by ϕ(y) = gy and ψ(z) = g(z) for every y ∈ Y and z ∈ Z, respectively. It is easy to see that both ϕ and ψ
are pointwise measurable, and that both ϕ and ψ satisfy condition (PI1).
By Fubini’s theorem, the
R pointwise integrals of ϕ and ψ exist and they are:
θϕ : Z →
R,
θ
(z)
=
ϕ
Y ϕ(y)(z) dµ1 (y) for every z ∈ Z and θψ : Y → R,
R
θψ (y) = Z ψ(z)(y) dµ2 (z) for every y ∈ Y , respectively. Moreover, using again
Fubini’s theorem, we obtain that both θϕ and θψ are µ2 - and µ1 -integrable,
respectively, and
Z
Z
Z
θϕ (z) dµ2 (z) =
θψ (y) dµ1 (y) =
g(y, z) d(µ1 ⊗ µ2 )(y, z). Z
Y
Y ×Z
Note that the definitions of ϕ and θϕ , and the measurability of θϕ as a
real-valued function defined on the measurable space (Z, Z) do not depend on
µ2 . Therefore, by Remark A, the following result is obviously true.
Proposition 2.1. Let (Y, Y, µ) be a finite measure space, let (Z, Z) be
a measurable space, let Bb (Z) be the Banach lattice of all real-valued bounded
Z-measurable functions defined on Z, and let g : Y × Z → R be a bounded
function measurable with respect to the product σ-algebra Y ⊗ Z. Then the
function ϕ : Y → Bb (Z), ϕ(y) = gy for every y ∈ Y , where gy : Z →
R, gy (z) = g(y, z) for every z ∈ Z, is well-defined (in the sense that gy ∈
Bb (Z) for every y ∈ Y ) and is pointwise integrable. The pointwise integral
7
Vector integrals and a pointwise mean ergodic theorem
69
R
P- Y ϕ(y) dµ(y) (which is a real-valued Z-measurable bounded function on Z)
is defined by
Z
Z
Z
Pg(y, z) dµ(y)
gy (z) dµ(y) =
ϕ(y)(z) dµ(y) =
Y
Y
Y
R
for Revery z ∈ Z, where (as mentioned earlier) P- Y ϕ(y)(z) dµ(y) stands for
P- Y ϕ(y) dµ(y) (z).
R
We will call the pointwise integral P- Y ϕ(y) dµ(y) that appears in Proposition 2.1 a Bb (Z)-pointwise integral. The next result shows that such pointwise integrals appear quite naturally when dealing with transition functions.
Proposition 2.2. Let (Pt )t∈[0,+∞) be a transition function defined on a
locally compact separable metric space (X, d), suppose that (Pt )t∈[0,+∞) satisfies the s.m.a., and let ((St , Tt ))t∈[0,+∞) be the family of Markov pairs defined
by (Pt )t∈[0,+∞) . Then, for every f ∈ Bb (X) and every Lebesgue measurable
(L)
subset L of [0, +∞) of finite Lebesgue measure, the map ϕf
defined by
(L)
ϕf (t)
: L → Bb (X)
= St f for every t ∈ L is pointwise integrable.
Proof. We apply repeatedly Proposition 2.1 in the case where Y = L, µ
is the Lebesgue measure on Y , Z = X, Z = B(X), and g(f ) : L × X → R is
defined by g(f ) (t, x) = St f (x) for every (t, x) ∈ L × X, f ∈ Bb (X).
First, assume that f = 1A for some A ∈ B(X). Then, using (1.1), we
obtain that St 1A (x) = Pt (x, A) for every t ∈ L and x ∈ X. Since (Pt )t∈[0,+∞)
satisfies the s.m.a., the map (t, x) 7→ Pt (x, A), (t, x) ∈ L × X is jointly measurable with respect to t and x; as the map is also bounded, weR can apply
Proposition 2.1 in order to infer that the pointwise integral P- L St f (x) dt
exists in this case.
If f ∈ Bb (X) is a simple function; that is, if there exist n ∈ N, n
measurable subsets A1 , A2 , . . . , An of X, and n real numbers a1 , a2 , . . . , an
n
P
such that f =
ai 1Ai , then the map (t, x) 7→ St f (x), (t, x) ∈ L × X, is
i=1
jointly measurable with respect to t and x because St f (x) =
n
P
ai Pt (x, Ai )
i=1
for every (t, x) ∈ L × X, and the maps (t, x) 7→ Pt (x, Ai ), (t, x) ∈ L × X,
i = 1, 2, . . . , n are jointly measurable with respect to t and x. Thus,
R applying
Proposition 2.1 again, we obtain that the pointwise integral P- L St f (x) dt
exists in this case, as well.
Now, if f ∈ Bb (X), f ≥ 0, then there exists a sequence (fn )n∈N of
simple measurable functions that converges uniformly to f , and such that
0 ≤ fn ≤ f for every n ∈ N. Since St , t ≥ 0 are (positive) contractions
of Bb (X), the sequence of maps (t, x) 7→ St fn (x), (t, x) ∈ L × X, n ∈ N,
70
Radu Zaharopol
8
converges pointwise on L × X (actually, the sequence converges uniformly) to
the map (t, x) 7→ St f (x), (t, x) ∈ L×X. Accordingly, the map (t, x) 7→ St f (x),
(t, x) ∈ L × X, is jointly measurable in t and x, and is bounded.
Thus, we can
R
apply again Proposition 2.1 in order to conclude that P- L St f (x) dt exists.
Finally, if f ∈ Bb (X) is not necessarily a positive element of Bb (X), then
f = f + − f − , where f + = f ∨ 0 ∈ Bb (X), f − = (−f ) ∨ 0 ∈ Bb (X), f + ≥ 0,
f −R ≥ 0. Using the above
we obtain that both pointwise integrals
R discussion,
+
−
P- L St f (x) dt and P- L St f (x) dt exist, and the maps (t, x) 7→ St f + (x)
and (t, x) 7→ St f − (x) are both jointly measurable with respect to t and x,
and bounded; hence, the map (t, x) 7→ St f (x) = St f + (x) − St f − (x) is also
bounded and jointly measurable with respect to t and x. Using Proposition 2.1,
(L)
(L)
we obtain that the map ϕf : L → Bb (X) defined by ϕf (t) = St f for every
t ∈ L is pointwise integrable. Let J ⊆ R be a finite union of bounded intervals of R, and think of J as
the measure space defined by the σ-algebra of the Lebesgue measurable subsets
of J and the Lebesgue measure on J. Our goal now is to obtain sufficient
conditions under which a function ϕ : J → Bb (X) is pointwise integrable,
is Bochner integrable in the sense of Appendix E of Cohn [1], and has the
property that the pointwise and Bochner integrals are equal.
R If ϕ is Bochner integrable in the sense of Cohn [1], we will use the notation
B- J ϕ(t) dt for the Bochner integral of ϕ on J.
Theorem 2.3. Let ϕ : J → Bb (X) be a continuous bounded function,
and assume that the range of ϕ is included in Cb (X). Then the following
assertions are true:
(a) ϕ is pointwise integrable;
(b) ϕ is Bochner integrable in the sense of Appendix E of Cohn [1];
(c) the pointwise and the Bochner integrals of ϕ are equal.
Proof. (a) Let g : J × X → R be defined by g(t, x) = ϕ(t)(x) for every
(t, x) ∈ J × X.
We first note that g is jointly continuous with respect to t and x. Indeed,
let ((tn , xn ))n∈N be a convergent sequence of elements of J × X, set (t, x) =
lim (tn , xn ), and let ε ∈ R, ε > 0. Since ϕ is continuous and (tn )n∈N converges
n→∞
to t, there exists mε ∈ N such that kϕ(tn ) − ϕ(t)k < 2ε for every n ≥ mε .
Since ϕ(t) ∈ Cb (X) and (xn )n∈N converges to x, there exists nε ∈ N such that
|g(t, xn ) − g(t, x)| < 2ε for every n ≥ nε . Clearly, we may choose nε such that
nε ≥ mε . Then, for every n ∈ N, n ≥ nε , we have
|g(tn , xn ) − g(t, x)| ≤ |g(tn , xn ) − g(t, xn )| + |g(t, xn ) − g(t, x)| ≤
≤ kϕ(tn ) − ϕ(t)k + |g(t, xn ) − g(t, x)| <
ε ε
+ = ε.
2 2
9
Vector integrals and a pointwise mean ergodic theorem
71
Since g is jointly continuous with respect to t and x, it also is jointly
measurable with respect to t and x. Clearly, g is bounded (because of the
boundedness of ϕ); since J has finite Lebesgue measure, we can apply Proposition 2.1 in order to conclude that ϕ is pointwise integrable.
(b) Since ϕ is continuous, ϕ(J) is a separable subset of Bb (X) (that is,
ϕ has a separable range) and ϕ is measurable (i.e., ϕ−1 (E) is a Lebesgue
measurable subset of J whenever E is a Borel subset of Bb (X)). Accordingly,
ϕ is strongly measurable. Now, the map t 7→ kϕ(t)k, t ∈ J is continuous
(because |kϕ(t)k − kϕ(s)k| ≤ kϕ(t) − ϕ(s)k for every s ∈ J and t ∈ J, and
because ϕ is continuous) and bounded (because ϕ is bounded). Since J has
finite Lebesgue measure, the map t 7→ kϕ(t)k, t ∈ J is integrable. Accordingly,
ϕ is Bochner integrable in the sense of Appendix E of Cohn [1].
n
P
(c) Note that if ψ : J → Bb (X) is of the form ψ(t) =
1Ai (t)fi , t ∈ J,
i=1
for some n ∈ N, n Lebesgue measurable subsets A1 , A2 , . . . , An of J and n elements (real-valued bounded measurable functions defined on X) f1 , f2 , . . . , fn
of Bb (X), then ψ is obviously strongly measurable (such a function ψ is called
a strongly measurable simple function) and Bochner integrable in the sense
of Cohn (because J has finite Lebesgue measure), and, by definition, the
n
R
P
Bochner integral in the sense of Cohn [1] of ψ is B- J ψ(t) dt =
λ(Ai )fi ,
i=1
where λ is the Lebesgue measure (on J). Also, ψ is pointwise integrable ben
P
cause for every x ∈ X the function t 7→
1Ai (t)fi (x), t ∈ J, is Lebesgue
integrable since t 7→
n
P
i=1
1Ai (t)fi (x), t ∈ J, is a simple measurable real-
i=1
valued map, and J has finite Lebesgue measure; moreover, the pointwise
n
R
P
λ(Ai )fi (x) for every x ∈ X; hence,
integral of ψ is P- J ψ(t) dt (x) =
i=1
R
R
P- J ψ(t) dt = B- J ψ(t) dt.
Since ϕ is Bochner integrable in the sense of Cohn [1] (as we proved at
(b)), using Proposition E2, p. 351 of Cohn [1], we obtain that there exists a
sequence (ψk )k∈N of strongly measurable simple functions, ψk : J → Bb (X)
for every k ∈ N, such that (ψk (t))k∈N converges uniformly on X (that is, in the
norm topology of Bb (X)) to ϕ(t) for every t ∈ J, such that kψk (t)k ≤ kϕ(t)k
for every k ∈ N and t ∈ J, and such thatR the sequence
of Bochner integrals
(which do exist as pointed out above) B- J ψk (t) dt k∈N converges uniformly
R
on X to B- J ϕ(t) dt.
(x)
(x)
(x)
Now, for every x ∈ X, the sequence (uk )k∈N , uk : J → R, uk (t) =
ψk (t)(x) for every t ∈ J and k ∈ N, converges pointwise to u(x) : J → R,
72
Radu Zaharopol
10
u(x) (t) = ϕ(t)(x) for every t ∈ J (because (ψk (t))k∈N converges uniformly
on X to ϕ(t) for every t ∈ J), so we can apply the dominated conver(x)
genceR theorem to the
sequence (uk )k∈N in
infer that the sequence
R order to
P- J ψk (t) dt (x) k∈N converges to P- J ϕ(t) dt (x). Thus, the sequence
R
R
P- J ψk (t) dt k∈N converges pointwise on X to P- J ϕ(t) dt.
R
R
Since (as shown in the first part
R of the proof) B- J ψk (t) dt = P- J ψk (t) dt
for every k ∈ N, and since B- J ψk (t) dt k∈N converges uniformly on X to
R
R
R
B- J ϕ(t) dt, we conclude that B- J ϕ(t) dt = P- J ϕ(t) dt. It can be shown that the integral discussed in Dunford and Schwartz [3]
is an extension of the classical Bochner integral as presented in Appendix E of
Cohn [1]; that is, if ζ : J → Bb (X) is Bochner integrable in the sense of Cohn
[1], then ζ is integrable in the sense of Dunford and Schwartz [3] and the two
integrals are equal. If we apply Theorem 2.3 to the integrals that appear in
Theorem 1.1 and Proposition 2.2, we obtain the following result.
Corollary 2.4. Let (Pt )t∈[0,+∞) be a Feller transition function defined
on a locally compact separable metric space (X, d), let ((St , Tt ))t∈[0,+∞) be
the family of Markov-Feller pairs defined by (Pt )t∈[0,+∞) , and assume that
(Pt )t∈[0,+∞) is C0 (X)-strongly continuous. If J is a finite union of finite length
(J)
subintervals of [0, +∞) and if f ∈ C0 (X), then the map ϕf
: J → Bb (X)
(J)
ϕf (t)
defined by
= St f for every t ∈ J is pointwise integrable, and Bochner
integrable in the sense of Cohn [1] and of Dunford and Schwartz [3]. Moreover,
the pointwise integral, and the two Bochner integrals in the sense of Cohn [1]
and in the sense of Dunford and Schwartz [3] are equal.
(J)
The proof of the corollary is obvious since the map ϕf that appears in
the corollary satisfies the conditions of Theorem 2.3.
The
corollary is useful because it allows us to think of integrals of the
R
form J St f dt, f ∈ C0 (X), as Bochner integrals in the sense of Dunford and
Schwartz; hence, we can use various results of [3] to study these integrals.
We now turn our attention to another type of pointwise integral that we
call the M(X)-pointwise integral.
Let (Ω, Σ, ν) be a measure space, let ϕ : Ω → M(X) be a function, and let
us think of M(X) as a Banach lattice of real-valued functions defined on B(X).
Then it makes sense to consider the definitions stated at the beginning of this
section in the present setting. Thus, we say that ϕ is pointwise measurable,
(ϕ)
or M(X)-pointwise measurable, if, for every A ∈ B(X), the function ψA :
(ϕ)
Ω → R defined by ψA (t) = ϕ(t)(A) for every t ∈ Ω is measurable, and we
say that ϕ is pointwise integrable, or M(X)-pointwise integrable if, for every
11
Vector integrals and a pointwise mean ergodic theorem
73
(ϕ)
A ∈ B(X), the function ψA is integrable and the map I : B(X) → R defined
R (ϕ)
by I(A) = Ω ψA (t) dν(t) for every A ∈ B(X) belongs to M(X). If ϕ is
M(X)-pointwise integrable,
we call I the M(X)-pointwise integral of ϕ and we
R
use the notation P- Ω ϕ(t) dν(t) rather than I. Also, it is
R convenient to use the
notation ϕt for
ϕ(t),
t
∈
Ω,
and,
for
A
∈
B(X),
to
use
Ω ϕt (A) dν(t) for I(A)
R
rather than Ω ϕt dν(t) (A), which is formally correct, but is cumbersome
and less intuitive.
Naturally, the first concern when dealing with an integral is to find conditions for integrability. In the next result, we discuss such conditions. The
conditions are general enough for our purposes, and probably they are good
enough for most purposes.
Proposition 2.5. Let ϕ : Ω → M(X) be an M(X)-pointwise measurable function. If there exists an integrable function ρ : Ω → R such that
sup |ϕt (A)| ≤ ρ(t) for every t ∈ Ω, then ϕ is M(X)-pointwise integrable.
A∈B(X)
In particular, ϕ is M(X)-pointwise integrable whenever the measure ν is finite
and there exists M ∈ R such that |ϕt (A)| ≤ M for every t ∈ Ω and A ∈ B(X).
Proof. Let ϕ : Ω → M(X) and ρ : Ω → R be as in the statement.
(ϕ)
Let A ∈ B(X), and note that the function ψA : Ω → R defined by
(ϕ)
ψA (t) = ϕt (A) for every t ∈ Ω is measurable since ϕ is M(X)-pointwise
(ϕ)
(ϕ)
measurable. Since |ψA (t)| ≤ ρ(t) for every t ∈ Ω and ρ is integrable, ψA
is integrable for every A ∈ B(X). Thus, the map I : B(X) → R defined by
R (ϕ)
I(A) = Ω ψA (t) dν(t) for every A ∈ B(X) is well-defined; accordingly, in
order to prove that ϕ is M(X)-pointwise integrable, we have to show that I
belongs to M(X). Since I(∅) = 0, in order to prove that I ∈ M(X), we only
have to prove that I is σ-additive.
To this end, let (An )n∈N be a sequence of mutually disjoint measurable
subsets of X. We have to prove that
∞
X
I ∪∞
A
=
I(An );
n=1 n
(2.1)
n=1
that is, that the series
∞
P
I(An ) is convergent and the above equality holds true.
n=1
Since ϕt is a signed measure for every t ∈ Ω, we have
I
∪ni=1
Z
Ai =
Ω
ϕt (∪ni=1 Ai )
dν(t) =
n Z
X
i=1
Ω
ϕt (Ai ) dν(t) =
n
X
i=1
I(Ai )
74
Radu Zaharopol
for every n ∈ N; thus, if lim I (∪ni=1 Ai ) exists, then the series
n→∞
convergent, and
∞
P
n=1
12
∞
P
I(An ) is
n=1
I(An ) = lim I (∪ni=1 Ai ). This means that in order to
n→∞
prove (2.1) it is enough to prove that lim I (∪ni=1 Ai ) exists and
n→∞
(2.2)
lim I (∪ni=1 Ai )
n→∞
(ϕ)
Now, note that ψ∪n
i=1 Ai
= I (∪∞
n=1 An ) .
is a sequence of integrable functions that
(ϕ)
ψ n
∪
Ai (t) ≤ ρ(t) for every n ∈ N
n∈N
(ϕ)
ψ∪∞ An ,
n=1
converges everywhere on Ω to
i=1
(ϕ)
and t ∈ Ω, and ψ∪∞ An (t) ≤ ρ(t) for every t ∈ Ω. Thus, we can apply the
n=1
(ϕ)
Lebesgue dominated convergence theorem to the sequence ψ∪n Ai n∈N and
i=1
R (ϕ)
R (ϕ)
conclude that lim Ω ψ∪n Ai (t) dν(t) = Ω ψ∪∞ An (t) dν(t).
n=1
i=1
n→∞
R (ϕ)
n
Thus, (2.2) is true since I (∪i=1 Ai ) = Ω ψ∪n Ai (t) dν(t) for every n ∈ N
i=1
R (ϕ)
and I (∪∞
A
)
=
ψ
(t)
dν(t).
n=1 n
Ω ∪∞
n=1 An
We have therefore proved that I ∈ M(X), so ϕ is pointwise integrable.
If ν is a finite measure, and there exists M ∈ R such that |ϕt (A)| ≤ M
for every A ∈ B(X) and t ∈ Ω, then ϕ is pointwise integrable because we can
use the above arguments applied to ϕ and ρ = M · 1Ω . The next result consists of an application of Proposition 2.5 to answer
a concern raised in Section 1; namely, under fairly general conditions,
we can
Rs
use the corollary to assign a meaning to integrals of the form 0 Tt µ dt, where
s ∈ [0, +∞), µ ∈ M(X), and (Tt )t∈[0,+∞) is the semigroup of operators on
M(X) generated by a transition probability.
Corollary 2.6. Let (Pt )t∈[0,+∞) be a transition function defined on
(X, d), assume that (Pt )t∈[0,+∞) satisfies the s.m.a., let ((St , Tt ))t∈[0,+∞) be
the family of Markov pairs defined by (Pt )t∈[0,+∞) , and let L be a Lebesgue
measurable subset of [0, +∞) that has finite Lebesgue measure. Then, for every µ ∈ M(X), the map t 7→ Tt µ, t ∈ L, is M(X)-pointwise integrable.
Proof. Let µ ∈ M(X), and let ϕ : L → M(X), ϕt = Tt µ for every t ∈ L.
We have to prove that ϕ is M(X)-pointwise integrable.
We first note that ϕ is M(X)-pointwise measurable. Indeed, let A ∈
B(X), and note that, since (Pt )t∈[0,+∞) satisfies the s.m.a., we can apply
Proposition 5.2.1-(a), p. 159 of Cohn [1] to the measure spaces (X, B(X), µ)
and (L, L(L), λ), where L(L) is the σ-algebra of all Lebesgue measurable subsets of L and λ is the Lebesgue measure, and to the map R(t, x) 7→ Pt (x, A),
(t, x) ∈ L×X, in order to obtain that the map t 7→ ϕt (A) = X Pt (x, A) dµ(x),
13
Vector integrals and a pointwise mean ergodic theorem
75
t ∈ L, is measurable; accordingly, the M(X)-pointwise measurability of ϕ is
established.
Now, note that |ϕt (A)| ≤ |ϕt |(A) ≤ |ϕt |(X) = |Tt µ|(X) ≤ Tt |µ|(X) =
kTt |µ|k = k|µ|k = kµk for every t ∈ L and A ∈ B(X). Since L has finite
Lebesgue measure, using Proposition 2.5 we obtain that ϕ is M(X)-pointwise
integrable. In the next result
R we discuss a useful relationship between theR Bb (X)pointwise integral P- L St f dt and the M(X)-pointwise integral P- L Tt µ dt
in the case in which we deal with a family ((St , Tt ))t∈[0,+∞) of Markov pairs
defined by a transition function (Pt )t∈[0,+∞) that satisfies the s.m.a., f ∈
Bb (X), µ ∈ M(X), and L is a Lebesgue measurable subset of [0, +∞) that
has finite Lebesgue measure.
Proposition 2.7. Let (Pt )t∈[0,+∞) be a transition function defined on
(X, d), assume that (Pt )t∈[0,+∞) satisfies the s.m.a., let ((St , Tt ))t∈[0,+∞) be
the family of Markov pairs defined by (Pt )t∈[0,+∞) , and let L be a Lebesgue
measurable subset of [0, +∞) that has finite Lebesgue measure. Then
Z
Z
(2.3)
P- St f dt, µ = f, P- Tt µ dt
L
L
for every f ∈ Bb (X) and µ ∈ M(X).
Note that (2.3) Rmakes sense because, by Proposition 2.2, the Bb (X)pointwise
integral P- L St f dt exists while the M(X)-pointwise integral PR
T
µ
dt
exists
by Corollary 2.6.
L t
Proof. We first note that it is easy to see that it is enough to prove the
proposition under the assumption that µ ≥ 0. Thus, assume that µ ≥ 0.
We will prove that (2.3) holds true in three steps: first for f = 1A
for some A ∈ B(X), then for the case when f is a simple B(X)-measurable
function and, finally for the general case when f ∈ Bb (X) is not necessarily a
simple function.
Step 1. Assume that f = 1A for some A ∈ B(X). The fact that
(Pt )t∈[0,+∞) satisfies the s.m.a. allows us to apply Proposition 5.2.1-(b), p. 159
of Cohn [1], and we obtain that
Z Z
Z
P- St 1A dt, µ =
P- St 1A (x) dt dµ(x) =
L
X
Z Z
=
X
L
Pt (x, A) dt dµ(x) =
Pt (x, A) dµ(x) dt =
L
L
X
Z
Z
= P- Tt µ(A) dt = 1A , P- Tt µ dt .
L
Z Z
L
76
Radu Zaharopol
14
Step 2. If f is a B(X)-measurable simple function (that is, if there exist m ∈ N, m measurable subsets A1 , A2 , . . . , Am of X and m real numbers
m
P
ai 1Ai ), then, using Step 1, we obtain easily
a1 , a2 , . . . , am such that f =
i=1
that (2.3) holds true for f .
Step 3. Assume now that f ∈ Bb (X). Clearly, it is enough to prove that
(2.3) is true under the assumption that f ≥ 0.
Thus, assume that f ≥ 0. Then there exists a monotone nondecreasing
sequence (fn )n∈N of real-valued positive simple B(X)-measurable functions
on
(fn )n∈N converges uniformly
X such
R that R to f . Therefore, the sequence
fn , P- L Tt µ dt n∈N converges to f, P- L Tt µ dt .
Since St is a positive contraction of Bb (X), the sequence (St fn )n∈N converges uniformly on X for every t. Thus, for every x ∈ X, the sequence of
(x)
(x)
(x)
positive functions (ξn )n∈N , ξn : L → R, ξn (t) = St fn (x) for every t ∈ L
and n ∈ N, is monotone nondecreasing and converges pointwise on L to ξ (x) :
L → R, ξ (x) (t) = St f (x) for every t ∈ L; accordingly, by the monotone conver
R (x)
R
gence theorem, the sequence L ξn (t) dt n∈N converges to L ξ (x) (t) dt. The
R (x)
R
fact that L ξn (t) dt n∈N converges to L ξ (x) (t) dt for every x ∈ X means
R
that the sequence of functions P- L St fn dt n∈N converges pointwise on X to
R
R
P- L St f dt. Since P- L St fn dt n∈N is a monotone nondecreasing sequence
of
R positive B(X)-measurable functions that converges pointwise on X to P(X, B(X), µ)
L St f dt, using the monotone convergence
R
R theorem in the space
we deduce that the sequence X P- L St fn (x) dt dµ(x) n∈N converges to
R
R
R
PS
f
(x)
dt
dµ(x);
that
is,
the
sequence
hPS
f
dt,
µi
cont
t
n
X
L
L
n∈N
R
verges to P- L St f dt,
µ .R
R
Since by Step
2 R P- L St fndt, µ
= Rfn , P- LTt µ dt for every n ∈ N,
we conclude that P- L St f dt, µ = f, P- L Tt µ dt . If L is an interval with endpoints a and b, a ≤ b, and if ϕ is a Bb (X)valued function on L, then we denote the Bb (X)-pointwise integral and the
Rb
Rb
Bochner integral of ϕ by P- a ϕt dt and B- a ϕt dt, respectively; similarly, if
Rb
ϕ is an M(X)-valued function on L, then P- a ϕt dt stands for the M(X)pointwise integral of ϕ.
In Section 1 we pointed out that,Rin order to obtain a weak* mean ergodic
t
theorem dealing with the averages 1t 0 Ts µ ds, t > 0, where µ ∈ M(X) and
(Tt )t∈[0,+∞) is the semigroup of operators defined on M(X) by a transition
Rt
function, we have to assign a meaning to the integrals 0 Ts µ ds, t ≥ 0. In
view of our discussion so far, we define these integrals to be M(X)-pointwise
integrals; by Corollary 2.6, the integrals exist whenever the transition function
15
Vector integrals and a pointwise mean ergodic theorem
77
defining (Tt )t∈[0,+∞) satisfies the s.m.a. Naturally, the integrals are denoted
Rt
P- 0 Ts µ ds, t ≥ 0.
As usual, given a function ϕ : (0, +∞) → M(X), we say that ϕ converges
in the weak* topology of M(X) as t → +∞ if there exists µ ∈ M(X) such
that the limit lim hf, ϕ(t)i exists and is equal to hf, µi for every f ∈ C0 (X);
t→+∞
in such a case we also say that the limit of ϕ in the weak* topology of M(X)
exists as t → +∞ and is equal to µ. We call µ the weak* limit of ϕ as t → +∞
and we denote µ by w*- lim ϕ(t).
t→+∞
Our discussion so far allows us to state the weak* mean ergodic theorem
mentioned in Section 1 that can be used to prove the pointwise mean ergodic
theorem for transition functions (Theorem 1.1).
Theorem 2.8 (Weak* Mean Ergodic Theorem for Transition Functions).
Let (Pt )t∈[0,+∞) be a Feller transition function that is C0 (X)-strongly continuous and C0 (X)-equicontinuous. Then, for every µ ∈ M(X), the limit
Rt
w*- lim 1t P- 0 Ts µ ds exists and is an invariant element for (Tt )t∈[0,+∞) in
t→+∞
Rt
the sense that if µ∗ = w*- lim 1t P- 0 Ts µ ds, then Tt µ∗ = µ∗ for every t ∈
t→+∞
[0, +∞); if µ ≥ 0, then µ∗ ≥ 0, as well.
We will not prove the theorem here; a proof will be given in the book
that we mentioned earlier that we are currently writing.
Note that if µ = δx , x ∈ X and L = [0, s], s ∈ [0, +∞), then (2.3)
becomes
Z s
Z s
Z s
(2.4)
St f (x) dt = PSt f dt, δx = f, PTt δx dt
0
0
0
for every f ∈ Bb (X). Using (2.4) we can easily see that, indeed, as mentioned
before stating Theorem 2.8, the truth of the theorem implies that Theorem 1.1
holds true. Equation (2.3) of Proposition 2.7 is also used in the proof of Theorem 2.8 in order to “transfer” the study of M(X)-pointwise integrals to Bb (X)pointwise integrals; since under the conditions of Theorem 2.8, Corollary 2.4
can be used, we obtain that the Bb (X)-pointwise integrals under consideration
are also Bochner integrals, so we can use the powerful machinery of Dunford
and Schwartz [3], and then, using (2.3) again, we can “return” to M(X)pointwise integrals.
Acknowledgements. Many thanks are due to Nicolae Dinculeanu for an exchange
of e-mails that has been extremely helpful in writing the paper, and to Jan van Casteren for comments on a previous version of the article that have been very useful in
improving the work.
78
Radu Zaharopol
16
REFERENCES
[1] D.L. Cohn, Measure Theory. Birkhäuser, Boston, 1980.
[2] N. Dinculeanu, Vector Integration and Stochastic Integration in Banach Spaces. Wiley,
New York, 2000.
[3] N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory. Wiley, New
York, 1988.
[4] H. Heyer, Probability Measures on Locally Compact Groups. Springer, Berlin–Heidelberg–
New York, 1977.
[5] H. H. Schaefer, Banach Lattices and Positive Operators. Springer, New York–Heidelberg–
Berlin, 1974.
[6] R. Zaharopol, Invariant Probabilities of Markov-Feller Operators and Their Supports.
Birkhäuser, Basel, 2005.
[7] R. Zaharopol, Invariant probabilities of convolution operators. Rev. Roumaine Math.
Pures Appl. 50 (2005), 387–405.
[8] R. Zaharopol, An ergodic decomposition defined by transition probabilities. Submitted.
Received 30 March 2007
Mathematical Reviews
416 Fourth Street
Ann Arbor, MI 48103, USA
[email protected]