ZERO-ONE LAWS FOR EXTREME MEASURES
by
Randall M. King
Department of Statistics
University of North Carolina at ~aapel Hill
and
Vlilliam E. Vilkinson
Depca'tment of Mathematics
Duke University
Institute of Statistics Himeo Series No. 952
Septenber, 1974
ZERO-ONE LAWS FOR EXTREME MEASURES
BY
".
P.andall M. .,lng
Deparotment of Statistics
University of N01"th Carolina at ChapeZ HiZZ
and
William E. Wilkinson
Deparotment of Mathematias
Dulw University
P,BSTRACT
The notion of an
extre~e
neasure is extended to
and useful characterizations of such
r~easures
extrene probability Deasure on the Borel sets
any subgroup in the completion of
1.
oo
B
certain linear spaces
are derived.
oo
B
of
\oJith respect to
00
E
J..1
If
J..1
is an
i t is shown thet
has measure
o or
This result leads, by means of an extreme-preserving napping theorel"!, to
an analogous result on the Borel a-field of an arbitrary separable Hilbert
space.
A series expansion for weakly continuous stochastic processes rlakes it
possible to obtain this 0-1 dichotomy for the space
CeT)
of real-valued con-
tinuous functions defined on an arbitrary Lebesgue measurable set T , witn
the a-field of cylinder sets.
In each setting, these results include as
special cases the previously known results for Gaussian measures.
-2-
1.
INTRODUCTION.
THEOREM.
Let
In 1970, Kallianpur [10] proved the following zero-one law:
X be a linear space of real-valuec functions defined on a com-
plete metric space T and let
sets.
~
If
of
G
be the a-field generated by the cylinder
is any zero-mean Gaussian ~easure on
ous·· covariance function, then
group
SeX)
~(G)
=0
or
(X, B(X))
with a continu-
for any B(X)-measurable sub-
I
X.
At about the same
ti~e, J~1ison
and Grey [9] obtained the same zero-one
result for completion measurable subgroups of the probability space
00
00
(R , B ,
~)
~
, where
countable product of
is the product normalized Gaussian
t~le
rea 1 line
00
R
with Borel a-field
~easure
on the
BOO • They then
applied this result to prove the Kallianpur theorem for completion measurable
subgroups in the case where
X = C(I)
~
is any zero-mean Gaussian measure on the space
of continuous functions on the unit interval.
Subsequently, Jain
[8] extended the more general result of Kallianpur to completion measurable
subgroups, and Cambanis &id
r~sry
[3] demonstrated that the restrictions on
T
and the covariance function could be dropped.
More recently, work has been done on extending these results to certain
classes of
non-G~ussian
measures.
Zinn [15] has exhibited a class of non-
Gaussian measures for which the zero-one law is valid for completion measurable
00
subgroups of R .
Dudley and Kanter [6]
~1ave
proved that all measurable sub-
spaces of a vector space h8.ve measure zero or one
measure.
l'li
th respect to any stable
-3-
We further examine this problen by considering yet another class of
measures.
Specifically we use the notion of an extreme measure, a term intro-
duced by Skorokhod [13] in the context of Hilbert spaces.
t'le prove the zero00
one law for extreme probability measures on the space
00
B)
(R ,
[Theorem 3.2],
and we then use this result, together with the technique of Jamison and Orey
[9], to derive zero-one laws on other spaces as well.
Finally, it is shown
that in the contexts considered, Gaussian measures are extreme so that these
results are in fact extensions of the results previously cited.
2. EXTREME MEASURES. Let X be a vector space over the field R of real
numbers and let
B be a a-field of subsets of
the transformation
measurable.
X such that for every a EX,
a : (X, B) + (X, B) , defined by
a
A measure
=x
a a (x)
on X shall be assumed to mean, unless
~
stated, a finite measure on
For any measure
(X, B) .
~
on
and we shall also use
~
to represent the complete measure on
For two measures
~
and
v
on
(X, B)
absolutely continuous with respect to
set of all measures on
v«
~.
v
E - F
= {e
we write
and
~
L V
tV
E and
- f: e
v
F of
to mean
E , f
In particular, we define the difference set of
E,
E - a
lltV li
~
,
(X, B ) •
~
to mean
~
~
is
is mutually
is defined on the
and
X we define
F} •
€
(X, B) ,
if and only if both 1.1« v
€
and we write, for any a EX,
~«v
The equivalence relation
(X, B) by 1.1
For any two subsets
v
othen~ise
B with respect to
is the a-field obtained by forming the completion of
singular with respect to
+ a , is
for the set
D(E), by D(E)
E - {a}.
=E - E
For each measure
-4~
and element
a of
~a(B)
(2.1)
TIle element
a
of
= ~(B
-
a)
for each
of
B E 0
is denoted by M
~
~
J..o
on
~
if
such that
(X, B)
DEFINITION 2.1. For any subset L of X a measure ).J on X is
3
or extreme with respeat to
L
:I
if
ML
~ E
sum of two non-zero mutually singular measures in
is simply extreme if there is some subset
It is easy to see that if
then
M
V
c
I,!-extreme.
M .
L
Land
Consequently, if
).J
M •
L
are subsets of
is L-extreme and
In particular, then, if there is some su)set
~ E
M).J
L-extreme~
as the
X with
L
M , , then
L
L of
~
c
L' ,
).J
is
X for which
Consequent I y.'
we could just as well have defined a measure to be extreme if it is
in the sense of Definition 2.1.
•
).J is L-extreme.
is extreme with respect to the set
~
is L-extreme, then
c
vIe will say that
L of X for which
L'
L
~tten
cannot be
~a« ~
For any subset
~
M. denotes the set of measures
X,
by
is said to be an admissible translate of
X
The set of all admissible translates of
L
~a
X, define a new measure
M -extreme
~
In applications, however, it may be that the
existence of sufficient conditions for admissibility of a translate enables one
to utilize a proper subset
L of M , even thougIl an explicit characterization
~
of all admissible translates is not known.
venient to choose a subset
In addition, it is sometimes con-
L of admissible translates satisfying some struc-
tural property not necessarily satisfied by the set
DEFINITION 2.2. Let
or trivial if ).J(A)
).J
=a
be a measure on
or
~(Ac)
=a
•
Then
AE B
~
is
~-trivial
-5-
For any two subsets
rand
F of
F , E 6. F , is defined by E 6. F
and
~(E 6.
=0
F)
~(E)
,then
DEFINITION 2.3.
Let
= ~(F)
LeX.
=
X, the sywmetric difference of E
[(E n FC) u (E c n F)). Note that if
for any measure
Then A
B
€
~.
is L-invariant with respect to
~
= 0 for each
or simply L-invariant s if
~[A 6. (A -
LEMMA 2.1. Let A € BII
A is L-invariant with respect to
AC
a))
a
~,
L .
€
~
if and only if
~.
is L-invariant with respect to
PROOF. Using the fact that
= EC
(E - a)c
(2.2)
a
-
for any
= [Ac
[A 6. (A - a)]
it can be verified that
E eX, a
6.
X,
€
(A c - a)) , from which the result
follows.
LEMMA 2.2. A measure
~
eX, B)
on
Hence
~(Ac)
hence
a
€
L
~(A)
If
~[A 6.
=0
,
(A - a))
S
ll(A)
+
ll(A - a)
A is L-invariant by Lerama 2.1.
for every a
=0
=0
, then ll(A - a)
€
L.
Tnerefore,
lla(A)
~
€
~\
= ~ a (A) = 0
,so A is L-invariant.
then the preceding argument shows that
A is invariant and so
~-trivial
~
set is L-invariant with respect to
PROOF.
is in ML if and only if every
Conversely,
A
C
•
If
is L-invariant and
s~ppose
= ~(A - a) = ~(A) = 0
~(A)
=0
Thus
and
-6-
For
E
E
B , define the
II
~easure
II
E
on
(X, B)
B€ B.
for all
LEMMA 2.3. Let II be a measure in ML . If A is
II
A
then
ML .
€
PROOF.
It is straightforward to verify that, for any A , B E B , a
(2.3)
[(A n B) - a] 6 [A n (B - a)]
C
ll[(A n B) - a]
(2.4)
B
€
= ll[A
.
llA(B) = 0
B be such that
X,
€
[A 6 (A - a)] .
In particular, then, if A is L-invariant and a
Let
L-invariant~
L ,
€
n (B - a)]
Then
II [(A n B)
- a] = 0 for each
Thus by (2.4) , (/') a (B) = II [A n (B - a)] = II [(A n B) - a] ,
L
A
A
A
= 0 ) demonstrating that (ll) «~l
for every a € L ; that is, II € M
a
L
a
€
L as
M .
].J €
THEOREM 2.1.
A
measure
j.J is L-extreme if and only if the families of L-
invariant sets and j.J-triviaZ sets coinaide.
PROOF. Suppose
].J
is L-extreme.
of every j.J-trivial set follows from
In particular,
Lej~a
2.2.
II
€
ML so the L-invariance
To show that every L-invariant
set is ll-trivial, let A be L-invariant; by Lemmas 2.1 and 2.3,
are both in
My • Thus
'-'
].J
= j.JA
11-,~
and
C
II
A
C
+ II
A
is a decomposition of
II
into two
mutually singular measures in ML
But II is extreme so that either
C
A
j.J
is identically 0 ; that is, A ~s ll-trivial.
j.J A or
-7-
Conversely, suppose
of sets cannot coincide if
case
~ ~
ML
~
By Lemma 2.2, the two families
it therefore suffices to consider the
ML . Hence there exist non-zero mutually singular measures
in M
L such that
~2
~1
~ E
is not L-extreme.
~
~
= }Jl
+
Then there is a set A
}J2 .
E
B
~
~l
and
such that
C
A
=~ ,
measures.
~
=
~
A
Since
for every a
E
is not
~-trivial
= 0 and. ~;-\
I/(Ac)
as
~l
are both non-zero
M ' it follo\'/s (using (2.2)) that,
L
E:
L ,
Likewise,
~[AC n (A - a)]
~[A ~
and hence
]J-trivial.
eA - a)]
= ~A
c
(A- a)
=
= O. Thus A is an L-invariant set which is not
This completes the proof.
Since equivalent measures have the same set of admissible translates, a
corollary of the preceding theoren is that measures equivalent to an extreme
measure are themselves
3. A ZERO-ONE LAW FOR
x
= (xl'
x2 ' •.. )
extre~e.
R~.
Let
of real numbers.
~
R
be the set of all sequences
We will use the characterization of
extreme measures given by Theorelll 2.1 to show that I!\easurable subgroups have
measure
0 or
1 \vi th respect to any extreme probability measure.
This
result includes the result for Gaussian measures appearing in Jamison and 0rey
[9].
In addition, the result will be useful in deriving similar zero-one la\'!s
-8-
in other types of linear spaces.
The general setting throughout this section is as follows.
00
the a-field of Borel sets in
a value of
oo
be
B
be the element of Roo which has
Let
R
Let
1 in the kth coordinate and
0 elsewhere.
For every n, let
Rn be the subspace of Roo sp~nned by {E , ... ,En} and let Rn be the subl
space spanned by {E k : k > n}. Let Bn > Bn be the Borel a-fields induced by
the subspace topologies on Rand Rn , respectively.
n
LHlI'''lA 3. 1•
for all a
If
(13, p. 562]
R
n
€
is a probabiU,ty measure on Rn and
V
then
!)
is
mn
Lebesgue measure on
v «v
a
(R
n
,B)
n
Roo can be identified with R x Rn in a natural Eanner.
n
oo
B"" = B x rfl as B is generated by finite-di~ensiona1 rectangles.
n
Note th~t
Moreover,
Thus we can consider
n ' Bn )
(R
n
n
and
(Roo
, B)
(R
as the product of the measurable spaces
Boo)
and we will make this identification for the
remainder of the section.
For each
Rn-section of
n
~
1 and
n
x.
n
R
€
and each subset
of
00
R , we define the
E determined by y by
EY
Similarly, for
y
X E
= {x
€
Rn : (x,y)
Rn ,we denote by
€ E} •
determined by
the Rn -section of
ine following result is a modification of a result stated only in the
context of Hilbert spaces in Skorokhod [13].
THEOREM 3.1. Let n
such that
ReM
n
~
~
I
and suppose
Let
and
p
~
is a probabiZity measure on
be the projections of
~
onto
00
(R
oc·
,B~.
(R , B )
n
n
-9-
and
n
, If) , respective?:y.
(R
PROOF.
A~ m
Then
n
and
Ax
~«
p •
A is defined by
A(Bn ) = ~(Bn
Rn)
x
for
Bn
Bn
E
There is a family of probability measures
such that
~
n
(x , B )
X E
is Bn -~easurable in
Rn } defined on
for each
x
n
B
E
rfl
and
such that
~(Bn
(3.1)
x
n
B)
= J ~(x,
Bn)A(dx)
B
n
for all
13
n
Bn ,B
E
n
E
n
B
(Loeve (11; p ..,36]).
From (3.1) we obtain the
representation
= J ~(x,
~(B)
(3.2)
R
n
Bn)A(dx)
x
Consider now the mea5ures
B
11
n
x B
EO
B
n
x
~~
a.
for
on
a
Rn
E
For any
Tfl
~a(Bn
(3.3)
x
n
B)
=
fB ~(x
-
a ,
Bn)Aa(dx) .
n
By hypothesis
1·
a
«~
and
is separable.
Fence, by Theorem 4 of Skoro khod
[13], the following two conditions are satisfied for each
A «A
(3.4)
(3.5)
a
~(x
- a , 0)« ll(X,·) a.s. [Aa(dx)] .
a
E
Rn
-10-
Let
Condition (3.4), together with LemMa 3.1, yields
Bn
Then, for
= dAm (x)
---d
n
n
B ,
€
p(Bn )
(3.6)
f(x)
= ll(Rn
x
il
B)
=
I
ll(X , Bil)A(dx) .
R
n
~ denote the measure
Let
E
€
A x p
on
(R
)
n , Bn
x
(R
n
, Sn) .
Then, for
Ba> ,
= JR
p(En)f(a)m (da)
an
'\;
(3.7)
ll(E)
n
Using (3.6) in (3.7),
~(E) =
(3.8)
I
R
n
JR ll(X
,
E~)A(dx)A(da)
.
n
Note also that, by (3.2),
II
-x
=
(E)
I
R
for
ll(a, En )f(a)m (da)
a-x
n
x
€
R
n
n
By the invariance of mn , we then have
(3.9)
II
-x
(E)
= JR
II (x + a , Eil)f(x + a)m (da)
an
.
n
From (3.8) and (3.9) ,.,.e nay deduce that
E
€
Boo and
'\;
ll(E)
=0
Temporarily; fix
[A(dx)] .
R
n
a
as fol1m</s.
Gy (3.8), there is a A-nul1 set
•
I
(3.10)
'\;
ll« II
4N.
ll(X, En)A(dx)
a
=0
N
Suppose
such that for
.
From (3.10) it follows that
n
ll(X, E )
a
=0
a.s.
-11-
TI1US,
using (3.4) and (3.5),
=0
~(x + a , En)
(3.11)
a
a.s. [A
-a
(dx)].
hence
IRn ~(x
(3.12)
,En)A (dx) = 0
a -a
+ a
a
~
N.
N is Mn -null,
Since
~(x + a ,En)A (dx)m (da)
a -a
n
=
f
J
=f
Rn
=
ll(X
Rn Rn
IR ~(x
, ETI)f(x)m (dx)m (da)
ann
+
n
a , Ea
)f(x n
+ a)m (da)m (dx)
n
n
r ~ -x (E)mn (dx)
'R
,
n
where the last equality follows from (3.9).
But
II
ll(E)
= 0 and so II «
x
«ll
for each
LEMMA 3.2. Let
x €
'1J
II
=Ax
so that
II
x
~
II
interval~
=0
II -x (E)
for all
x €
a.e. mn (dx) .
R
n
Therefore,
p •
E be a Lebesgue measurable subset of
denote Lebesgue measure on
dimensionaZ
Rn
Thus
R
n
•
If
aentered at the
In
n
(E) > 0 ,
orig'in~
R
n
and let
m
n
then there exists an n-
oontained in
D(E)
The proof of this result can be obtained by a modification of the proof
for the special case
n
= I appearing in Halmos [7, p. 68].
-12-
Let
LO be the linear span of the set
THEOREM 3.2. Let
00
and Zet G E
be an Lo-extreme
~
be a group.
B~
PROOF. Suppose that
if
x
a
+
Bence
G.
E
G =G - a
Thus
for every
G is LO-invariant with respect to
G is ~-trivial; that is,
2.1,
ll(G) =
a
a
E
La
E
L '
O
R : (x,O)
E
E
n
so that
° or
I •
00
measure on Rn
~-measure
hence
As
y
In
"
before, let
=N
tion that
(x/N)
0
GO
n
;t
R
n
Go
E
n '
E
n
n
R , ~ (HY) > 0
in
n
GO
Eence
n
y
m (H )
n
n
B
~
m
n
,
(HY)
n n
that
Bn ) and
=0
is a group.
n
°=R
Gn
n
(A x p) (H)
=
°.
Hence
~(H)
, since
10 c G~ and
This contradicts the assunp-
x/N
E
= 0 for every Y E Rn .
(R n , Bn ) , respectively.
, it follO\'ls that ACHY)
n
G
R ,centered at the origin,
Since R c LO and ~ is Lo-extreme, it follows that Rn c M~
n
Theorem 3.1, ~«A x p , where A and p are the projections of II
(R
)
By Lemma. 3.2,
n
N sufficiently large so that
as
E
be Lebesgue
In
But this implies that
R , \ve may choose
n
x
00
G
00
and suppose that for some
such that
X E
n
B -measurable set H and a subset
o.
there exists an n-dimensional interval
if
=
GO
n
can be written as the disjoint union of a
of a B -measurable set of
=~
G 6 (G - a)
and consequently, by Theorem
~,
LO 4 G . Choose n ~ 1 such that
G} ~ R
CO is a subgroup of R
n
n
c:o
B)
,
G if and only
X E
Suppose now that
{x
00
(R
~-triviaZ.
TI1en) for every
La c G
so that
measure on
p~babiZity
G is
Then
= 1,2, ... l
k: k
{E
= 0
A tV I!1n
Furthermore,
for all
n
YE R
= 0 , and consequently
By
onto
..·1nce
C::'
But this implies
~
(G)
=0
-13-
We remark that Zinn [15, Corollary 1.3] has proved the same result for
measures
V satisfying the conditions
(i)
n
n
(R
n
,
n, ~. . here
for all
A «m
is the proj ection of
A
n
V
onto
13 )
n
is a product measure.
(ii) "ll
Theorem 3.1 shows that (i) is implied by our assumption that
L
O
c
M ' but
v
Condition (ii) is much nore restrictive.
We next establish the zero-one theorem of Jamison and Orey [9] for tI1e
00
00
Gaussian measures on
(R
THEOREM 3.3.
be a Gaussian produat meaSUI'e on
Let
V
as a special case of TI1eorem 3.2.
P
with covariance function
determined by
K.
K(j,k)
Then
V
by
R
= (xn )
= Cov(P j
H(K)
, where
00
on
k
x
= H(K)
00
, B ) •
l.l
(3.13)
l.l
00
B -measurabZe subgroups are v-trivial.
Define projection naps
[12J) that M
(R
00
is La-extreme and all
PROOF.
, B)
00
E
R
, P ).
k
It is well-known (Parzen
is the reproducing kernel Hilbert space
Furthermore, using the characterization of
Parzen [12J, it can be shown that
Thus
V
E
Mr
e:
n
E
H(K)
for every n
H(K)
given by
and hence
and it follows from Lemma 2.2 that every
l.l-
"'0
trivial set is LO-invariant.
Conversely, let
and
{P k : k
A be any LO-invariant set.
= 1,2, ... }
As
n
B
= o{pn+ 1
' Pn+ 2 , ... J
is a sequence of independent random variables, it
-14-
f.ollows from the zero-one lalfl' for the tail a-field generated by a sequence of
co
independent random variables that the a-field
trivial sets.
that A
CO
= BII
r
We will show that the invariant set
n
n STI
consists of
ll-
n=l lJ
A is lJ-trivial by showing
r .
€
By the LO-invariance of A , there is a lJ-null set
Na for each
a
€
LO
such that
(3.14)
For fixed
mean
n ,let
~
and covariance matrix the identity matrix
0
Rn with
co
R ,define
be the density of the normal distribution in
I
For
n
x
€
Since
f Q)!g(X)
- XA(x) IlJ(dx)
R
and Rn
C
LO
J
g(x) = XA(x)
Ma = NuN a u (N - a)
(3.16)
g(x) = XA(x) = xA(x
{a}
m
y) - XA(x)
+
g(x)
IlJ(dx)~(y)mn(dy)
= XA(x)
a.s. [1.l(dx)] • Hence
N such that
The set
Let
f colXA(x
R R
n
' it follows from (3.14) that
there is a lJ-null set
(3.15)
$
for
x
4N.
is also lJ-null for each
+ a)
= g(x
+
a)
be a dense subset of Rn and let
for
M=
x
a
€
LO and
4 Ma
u M
m=l am
For given
a
€
R
n
-15{a
let
: k
~
= 1.2 •... }
be a subsequence such that
00
straightforward to show that for
of
a
n
g (x
Thus
g(x
R • g(x
X E
= a .
111,
+
= limg(x
a)
k~
+
a
mk
= g(x)
)
is a constant function of a
+ a)
€
It is
is a continuous function
+ a)
This fact, together with (3.16). implies that for
R
€
lin a
k~
x
4 M•
.
almost surely; that is.
R
n
(3.17)
where
pn(x)
measUrable.
g
• x l ' x 2 •..• )
n+
n+
n
B -measurable by (3.17).
is
is Bn-measurable for
~
~
00
n Bn = r. T)~,us
n=l ~
is La-extreme and consequently. by Theoren 3.2.
"A "SOO,.,
= 1.2, ... now follows froM (3.15) so that
each n
A "is
= (0 •••.• 0
~-trivial.
Therefore
~
00
every B -measurable subgroup is
~
I I
~
~-trivial.
4. THE TRANSFORMATION THEOREM. The following theorem provides a means of
00
transition from the result in
R
given by Theorem 3.2 to analogous results
in other spaces.
THEOREM 4.1. Let
X be a real "linear space~
field of subsets of X and
(X.S)
and let
TI(L)
= La
map from
a subset of X
7T
X
~
Y onto 7T(Y), where
(R
if v is Lo-extreme.
00
of 7T
to
B(Y) = B n Y .
SIlO) • it foZZO/J)s that
(J-
be a measure on
containing
L
00
X into R
be a one-to-one linear map from
and the restriction TIl
induced measure on
Let
be a B--measurable subgroup of
Y
Let
L
B a tr'anslation invariant
such that
suoh that
Y is a B(Y) - BOO bimeasurable
If
~
'V
= llTI -1
denotes the
is L-extreme if and only
-16-
PROOF.
If
B ~ B , then
~(B)
(4.1)
= ~(B
~CB n
= ~I(B
Y)
= v(~(B
n Y)
Y)
n
ex>
E
B
and so
for all
n V))
B
E
B .
In particular,
~(B)
It follows that the
correspond under
= v(~(B))
~-trivial
~i
,
e1e~ents
for
E
B'
~
B
and hence
~j
~
BCY) .
ex>
ex>
B (Y) - B
is
bimeasurable.
v
~
establishes a one-to-one correspondence
L and elements
a of
E
Y and the v-trivial subsets of R
subsets of
Under the given hypotheses,
between
for all
a
l
= ~(a)
E
L , a'
of
LO '
Also note that
BO:> ,
-l(B! - a I) -_ ~-l(BI) - a
for
II
a
= ~(a)
E
LO .
Using the identity
(B n Y) - a - (B - a) n Y
(4.2)
if and only if v
it is easy to show that
Suppose then that
v
fA
€
L
o
but
Then n -1 (B')
a 1 = ~ (a)
€
Hence
L '
O
€
B
11
v
E
€
L
My
....0
is not Lr,-extrel'le.
IJ
By Lemma 2.2 and
which is Lo-invariant but not v-
Theorem 2.1, there is some set
trivial.
BE B , a
is not
~-trivial.
3ut for any
a
€
L ,
-17-
As
~-l(Bl)
is L-invariant but not ~-trivial, ~
On the other hand, suppose
set
B
E
~ E
~
ML but
is not L-extreme.
is not L-extreme.
which is L-invariant but not ~-trivial.
B
~
not v-trivial.
properties of
For any a'
E
LO ' a
= ~ -1 (a
l
)
There is a
~(B n
By (4.1),
ELand thus by (4.2)
Y)
is
a~d
the
,
~
This identity, together with (4.1), yields
Thus
~(B'n
Y)
is Lo-invariant but not v-trivial, so
COROLLARY 4.1. Let
~
be a probabiZity measure on
hypotheses of the preceding
Y of
B
~
~
X.» and a map
PROOF. If G E B
lJ
From Theorem 4.1,
Thus
v(~(G n V))
is not Lo-extreme.
(X,B)
satisfying the
theorem for some subset
from
-measumbZe subgroup of
v
X
X into
0:>
R
has measure
is a group, then
~(G
0
n
If
lJ
or
1.
Y)
L
of
is L-e:ctreme., then every
0:>
is a B -measurable subgroup.
v is an La-extreme probability measure on
is
a
or
1 by 'P-leorem 3.2.
X:; a subgroup
That
(R
0:>
0:>
,B).
G is lJ-trivial folloNS
from (4.1).
It should be noted that the subgroup
can be taken to be all of
Y of Theorem 4.1 and Corollary 4.1
X when it is convenient to do so.
-13-
5.
HILBERT SPACES. We first apply Theorem 4.1 to the case where X is a
separable Hilbert space.
THEOREM 5.1. Let
be a real separable Hilbert space with
H
Let
field of Borel sets of H.
and let
{e 1
(H, B(I-I))
measure on
and
the a-
be a complete orthonormal set in
}
.(
be the linear huZZ of
L
B(H)
If
{e }
k
G is any
H
is an L-extreme probability
II
B (H) -measurable suhgroup.• then
ll
G is
ll-triviaZ.
PROOF.
in H.
Let
Then any
II ' II
and
< • , 0>
x
denote the inner product and norm, respectivel;·;
H can be written in the form
E
00
=
x
I
<x
k=l
e > e
k
k
where the convergence is with respect to the Hilbert space norm and
II xl1
2
co
=
L <x
k=1
00
Furthermore, if
(x )
k
is any point in
R
such that
00
n
sequence
H
and
Yn
Xl.(
=
=L
k=l
<x
xke k
, ek >
x2 <
L
k
k=l
\
converges in H to some element
x =
00
00
L
k=l
Thus the mapping
from
1f
H
into
00
R
then the
~,ek
in
defined by
(5.1)
H onto
is a one-to-one linear transformation from
00
TI(H)
For each
k, TI(e k)
= E~
= {(xk )
I
E ROO:
so that
k=l
TI(L)
x~
< oo}
= La.
•
B(H)
is the smallest a-field
-19-
oo
B
measurable (M1mad [1, p. 100]) and
p
n
which the projections
measurable.
B
€
B(H)
1T
Thus
= <x
r n , defined by r n (x)
for which the functions
00
defined on
00
is
B
B(H) -
i~
R
, e > ,are
B(H) - B(R)
n
00
R
the smallest a-field in
= xn
by
measurable.
Moreover,
for
are
00
1T(H) c B
If
is of the form
B
(5.2)
= {x
€
H: <x , e >
n
~ t}
then
= {(xk)
1T(B)
Since
1T
00
R : xn
€
~ t} n
1T(H)
00
B •
€
B of H for which 1T(B)
is one-to-one, the set of all subsets
is a a-field containing all sets of the form (5.2).
B
€
so that
B(H)
11
on
II
(H, B(H))
continuous linear functional defined on
B(H) ,
ll).
Suppose
to be the unique element
I
(5.3)
Boo
€
for any
B (H)II
ill
in
is s~id to be Gaussian if every
H is a Gaussian random variable on
is Gaussian.
II
<x,y>ll(dx)
The me&1 element of
II
is defined
H satisfying
= <m,y>
, for every
y
€
H •
H
The covariance operator
S of
II
is the bounded linear operator defined on
H satisfying
(5.4)
J<x - m ,
H
B
H is ll-trivial.
A probability measure
(H ,
1T(B)
Thus, by Corollary 4.1, every
is bimeasurable.
measurable subgroup of
Thus
00
€
y> <x - m , pll(dx)
= <Sy
, z>
for all
y,Z
€
H .
-20-
If
f
(H, B(H) ,~)
x denotes the Gaussian randoM variable defined on
fX(Y)
= <x,y>
,then
covariance of
f
COROLLARY 5.1.
<x,m>
x and
Let
for all
~
f x and
is the expectation of
x,y
be Gauss,,/:an on
<Sx, y>
by
is the
H
EO:
~
'l'hen
(H, B(H)).
is extreme and
hence aU B (H) - measurable suhg!'oupS of H are ll-trivial.
~
PROOF.
Let
{e k } be a complete orthonormal set of eigenvectors of
co
corresponding eigenvalues
1ection
{e k } and. define the f'lap
rk ex) = <x ,
defined by
e >
k
'IT
"lith
L be the linear span of the ca1-
Let
{Ak}k=l
S
as in (5.1).
The random variables
are all Gaussian with !'lean
fk
<m, e > ; that is,
k
Moreover. using (5.4),
= A}<ok'J
Cov (fl.l, , f.)
J
Thus
{fk: k
= 1,2, ... } is
It fo11o\l1s that the set
where
o},
(J
k = j
- {ol
k
~
j
a family of independent Gaussian random variables.
of proj ections introduced in ti1e
{Pl: k = 1,2, •.. }
K
proof of Theoren 3.3 is a sequence of independent Gaussian random.variables
with respect to the induced measure
product Gaussian on
(R
co
00
, B ).
co
, B) .
~'IT
-1
H are
on
Bv Theorem 3.3,
It follows from Theorem 4.1 that
measurable subgroups of
6.
co
(R
v =
~-trivial
SPACES OF CONTINUOUS FUNCTIONS.
Let
~
co
co
B ) ; that is,
(R
v
v
is
is La-extreme on
is L-extreme and hence all
by Corollary 4.1.
X
= C(T) , the set of all continuous
functions defined on some Lebesgue measurable subset .,.,
of the real line.
Le·L.
-21-
SeX)
{x
be the a-field generated by IIcylinder" sets of the form
X: [x(t ), ... ,x(t )}
1
n
E
B}
n
E
Bn
all n-dimensional Borel sets
on
process
that
on
{r(t,x): t
B(X)
(x,
in
By the process
(X, B(X)).
for all
Rn
Let
{x(t): t
defined on
T}
~
n
E
(X
T}
t.1
1
T , 1
E
is precisely the smallest a-field nakin.c;
r(t,x)
for all
for the process
t
{x(t): t
T
E
E
T}
,~ill
desired zero-one theorCPl
We
~
n , and
He nean the stochastic
r(t,x)
S(X))
i
be a probability measure
~
by
E
~
B(X))
= x(t)
.
Note
a random variable
first develop the series representation
described by Cambanis and Masry [3].
The
then follow by arguments very similar to those
used in the Hilbert space case.
It follows from the assumption of path continuity that the process
{x(t): t
Let
E
is a
T}
K(s,t)
X , B(T)
x
x B(X)
= E[x(s)
~\Je can choose a neasure
\I
x(t)]
on
= fIxes)
[m]
and such that
Such
~ieasures
\I
x(t)~(dx)
.
n ,
''lith derivative
~\I(t)
~
....m
always exist by reans of a construction given in [3].
0 a.e.
Hence-
will always denote a measure with the properties given above.
such a measure
since
•
(T, BCT)) such that \I is absolutely con-
tinuous with respect to Lebesgue measure
forth,
-measurable process.
be the autocorrelation function of the process:
K(s,t)
(6.1)
(T
\I, it follmvs that
x(t)
E
L2 (T
,
B(T)
,
\I)
Fur
= L2 (\I) a.s. [~] ,
-22-
Moreover, by Schwarz's inequality and the fact that
K(s,t)
E
L2 (v x v)
operator
= 1.2 (T
K defined on
(6.2)
K(t,t)
x T , B(T) x B(T) • v x v)
Ll(T ,
B(T) , v) ,
Thus the integral
by
L (v)
2
(Kf)(t)
E
= fTK(S,t)f(S)V(dX)
is a Hilbert-Sc1mitt operator.
K
is self-adjoint and completely. continuous.
(See [2] for details.)
Let
H (x. T)
denote the subspace of
-
L"'l (J.I) = L? (X , B (X)
.:..
the mean square sense by
{x(t)}.
weakly continuous if for all
Since
~ E
The process
{x(t): t
H(x,T) , li:n E[x(s)
s+t
0
f;;]
, J.I)
E T}
spa:rm0c in
is said to be
= E[x(t) .,
f;;] •
(A~':hiezer
K is self-adjoint and completely continuous, it follows
and Glazman [2, p. 127]) that there is an orthonormal set of eigenfunctions
{fk } of the operator
K corresponding to nonzero eigenvalues
necessarily distinct - and
suc~
that
{f1•<. }
The orthogonal cOEplegent of the range of
h
(K\'0-
where
(0,0)
{x(t):
t
E
T}
,
of)
-k --
fh
I.,",
Kf)
1<. --
(h
u,
{A } - not
k
is complete in the range of
K is the null space of
K.
K, since
'oF)
-- 1\,,.
'('-0.. , "'1
f) '
1\1," 1,
l'
... \.
ol\.
(
denotes the inner product in
is weakly continuous,
From this it follows that the range of
in particular, the eigenfunctions
L (V). If the process
2
K(o,t) is continuous for each
t
E
T .
K consists of continuous functions;
f k are all rr.er::bers of
X
= C(T) .
-23-
The following result, while not appearing in quite this forn, is essentially contained in Caf.1banis and Hasry [3].
THEOREM 6.1. Let
.be a second order weakly continuous process
{x(t): t E T}
and define l'andom varialJles
r:-
"'1:
lJ".J'/
~k(x) = JTX(t)fk(t)V(dt)
(6.3)
Then each
~k
L2(~)
is in
spanned by
{~1) .
!.
ar~~
in
fact~
Moreover",
x(t)
H(x,T) =
H(~)
, the subspace of
has the repre8entation
(6.4)
where the convergence is in
THEOREM 6.2. Let
defined on
extreme and
PROOF.
L2(~)
{x(t): t E T}
X = C(T)
and in
L (V) a.s.
2
[~]
.
be a weakly continuous second order prooess
L be the linear span of {fk }. Lf ~ is L(X) -measurable subgroup" then G is ~··triv1:al.
and let
G i8 any
B
~
We use the representation
satisfying the conditions of
Theorer, 6.1, define a mapping
(~eveloped
TheorE;~!
'iT
4.1.
in
T~".eorerr
6.1 to define a mapping
Given the representation ((-).4) of
00
co
Beq)
into
(R
since t}:e paths
x (t)
are continuous.
fran
(X,
, B)
by
(6.5)
[v] , which i!'1plies thCl-t
xl
= x2
-24-
Let
Y be the set of all
By Theorem 6.1,
map.
~(Y)
SeX)
Since
=1
x in
Y is a group containing
measurable.
Thus
SeX)
SeX)
is an n-dimens ional Borel set.
B
n
Moreover,
in
7T'
TI
00
(Y)
E
B
E
as
= (f.] , f 1,e ) = 0.] 1.'.•
]
eac~
n
Thus
00
B(X) .- B
is
7T
Ik
if and only if
is bimeasurable, since if
~k (f .)
Since
x(t) measurable, the
B } ,
\Lk x 2
k
L (v) , and the latter is true if and only if
2
= 7TIY
TI
is generated hy sets of the form
1
where
L2 (v)
is a linear
is the smallest a-field mating
B = {x: [t:l( (x), ... '~1/ (x)]
'n
1
(6.6)
Land
is the sInallest a-field making each
representation (6.4) implies that
~k
X such that (6.4) converges in
,
xkf~(t)
<
= e:.J
, and thus
7T (L)
converges
In -tact,
00.
B is of the form (6.6) and
7T (f.)
J
measurable.
BeY
= Lo ' TIle
conclusion of the theorem now follows from Theorem 4.1.
~
We say that
{x(t): t
E
T}
is a Gaussian measure on
defined on
(X,
B(X))
(X, SeX))
PROOF.
Define
7T
00
~
as defined by (6.3), are in
Cov(~k
'
~j)
variables.
= AkO kj
(X
Sex))
B (X)-measurabZe subgroups of G are
as in (6.5).
'{~k}
~.
is Gaussian with respect to
COROLLARY 6. 1• If ~ is a Gaussian measure on
extreme and thus aZl
if the process
then
~
is L-
~-trivial.
By Theorem 6.1, the random variables
H(x,T)
Thus each
~
~k
is Gaussian.
~k'
Since
is a sequence of independent Gaussian random
Just as in the proof of Corollary 5.1, it follows that the induced
-25measure
'J =].l7T
La-extreme on
~easurable
-1
is product-GaussiH.Il on
00
and thus
(R
subgroups of
].l
(R
00
-aoo
,.;)) .
By Theorel!l 3.3,
is L-extre'T'e by Theorem 4.1.
'J
is
Hence all
X are \l-trivial by Corollary 4.1.
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[1]
[2]
[3]
A~~ad,
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Cambanis, S. &Masry, E. On the representation of weakly continuous
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[4]
Cambanis, S. &Rajput, B. S.
Trans. A>ne:r>. Math. Soc.
[5]
Cameron, R. H. & Gl'aves, R. E. Additive functiona1s on a space of continnous functions I. Trans. Amer. Math.Soc., 70, pp. 160-176 (1951).
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Dudley, R. M. &Kanter, M. Zero-one laws for stable measures.
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Ha1mos, P. R.
Measure Theory.
Zero-one laws for Gaussian processes.
149, pp. 199-211 (1970).
Proc.
Van Nostrand, Princeton, New Jersey (1950).
[8] Jain, N. C.
A zero-one law for Gaussian processes.
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Proo. Amer. Math. Soo.
[9] Jamison, B. &Orey, S.
[10]
(11]
[12]
Subgroups of sequences and paths. Froc. A~er.
Math. Soo. 24, pp. 739-744 (1970).
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Skorokhod, A. V. On admissible translations of measures in Hilbert space.
Th. Frob. Appl. ~,pp. 557-580 (1970).
[14]
Tucker, H. G. A Gra,duate Course in Probability.
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[15]
Zinn, Joel.
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Academic Press.
Zero-one laws for non-Gaussian neasures.
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A>ner. Math.