This research was paniaZZy suppo%"ted by the National Science Foundation
untie%" G%"ant GU-20S9.
1
A1,so~
Department of
MathematiC8~
Unive1'8ity of No%"th Ca%"olina
~ PasTRACT WIENER
rtAsuRE
BALRAK S. RA1PUT 1
Department of Statistics
Unive1'8ity of No%"th Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 713
September. 1970
ON ABSTRACT WIENER MEASURE.
BALRAM S. RAJPUT
n
Introduction.
space
An abstract Wiener measure is a o-extension in a Banach
(x. II· Il x ) of the canonical Gaussian cylinder measure Ux of a
real separable Hilbert space
on
X and
X such that the norm
X is dense in X in the norm
II' IIx·
II' Ilx
is measurable
Gross shoved [4, p. 39]
that there exists an abstract Wiener measure on &n'1 real separable Banach
space.
Sato showed [6, p. 73] that if
separable or reflexive Banach space
separable closed SUbspace
X of
u is a Gaussian measure on a real,
(X.
II· Ilx>
X such t'hat
then there exists a
u(X) = 1
a o-extension of the canonical Gaussian cylinder measure
separable Hilbert space
continuous on
X and
X.
X is dense in
is
uX ot a real
II' Iii:: II· 11X;X is
f in the norm II· IIx· The main
II· Iii is measurable (and not merely
Utilizing this and sato's result mentioned above, we
u on a real, separable or reflexive Banach
obtain that a Gau8sian measure
space X has a restriction
x on & closed,separable subspace
U
which i8 an ab8tract Wiener Masure.
i ot
X,
Using this result and a result of
Gross [4, p. 40]. we show that the subspace
P.
lIi'. u/I.
X such that the ~orm
purpose of this note is to prove that
continuous) on
and
X
of
X is the support ot
Specializing this result to real Hilbert spaces, we prove that the suppOrt
of a Gaussian measure on a real Hilbert space is the closure ot the linear
span of the eigenvectors corresponding to nonzero eigenvalues of the covariance
tlThls research va.e pa.rtlall.y supported by the lfatlonal Science 'Poundation .
under Grant 0tJ-2059.
2
operator of
Ito f5].
This result was recently reported in the literature by
l!.
We also obtain a necessary and sufficient condition for a
Gaussian meuure on a real, separable or reflexive Banach space to be an
abstract Wiener measure.
As
a corollary of this result, we get a necessary
and sutficient condition for a real, separable or reflexive Banach space
X to be the support of a Gaussian measure
In Section 2 t
fi~st
defined on
IJ
X.
we vill establish the notation and terminology
and then we will make a tev remarks about BODle of the results mentioned
above.
12
Definitiona
!S! 1>reliminaries.
The basic definitions, notation and properties that are used con-
e
Although most at the tolloYing tacts can be
sistently are given belay.
found in [4] and [6], we summarize them here for the sake of completeness
and rea47 reference.
Let
X be a real Banach space, X*
its conjugate space.
A CYlinder
subset of X is defined by
{x
wbere
~lt ••• '~
luclidean space
and
'llx
f:
X: (~l (x) , ••• t ~n (x»
.E
D}
(X· and D is a Borel subset at the n-dimensional
an.
The class
Ux
at all c11inder sets torms an algebra.
denote. the a-algebra generated b1 UX•
A probabilit 1 measure
lJ
on the meuurable epace (x,UX) is called Gaussian if tor everr t c X*,
t(x)
i8 .. Gaussian randaa variable with mean zero on the probabUit, apace
(x,~.JI).
It
JI
is finitely additive probabilit 1 meuure on
that
1
a
lJ{x E X: (x) < a) • - - - - - - Jexp[-
-
,I2;V'fff--
(X.Ux)
such
1 t2
'2 vnr]dt
..
".. ..
'
3
for every
t (X·
and real
a, then
i8 called Gaussian czlinder
\I
measure.
X be a real Hilbert space.
Let
measure
\IX
.
!b!. canonical Gaussian cYlinder
X is a finitely additive nonnegative set tunc'tion on
on
(X,U X) such that
1
PX{x ( X: t(x) ~ a} •
/2;
t (X*
for arq
~
and any real
canonical normal
(n,B,p)
F on
2
2]dt,
lit "X*
denotes the norm in
X* -
X is an equivalent class of
F) such that tor every
t (X* the distri-
lit II~.
i8 Gaussian with mean zero and variance
Two
'1"2 are equivalent if tor every finite number of elements
maps
tl'·· •• ~
£ >
lit IIx- -where II· II X*
distribution
(depending on
bution of F(t)
to~
J exp[-
X* to the space of random variables over a probability
linear maps from
space
~
a,
t2
a
(X- 'the joint distribution of . 'J (tl ) ... "'j (tn )
.1. 1,2. A norm
II· lion
is the same
X is called measurable if tor every
0 there exists a finite dimensional projection Po
(depending on
such that for every finite dimensional projection P orthogonal to
e)
Po
we have
l'X{x .: X:
IIPx II
>
d
< e.
The two concepts - canonical Gaussian cylinder measure and canonical
normal weak distribution - are indeed equivalent [4, p. 32].
Since we will
have oocas.ion to use both, we have included them here.
Let
and
be a continuous nora on a real separable Hilbert space
X be the Banach space obtained by the completion of
II· II.
tit .
II- II
Through the natural embedding
X-
can be considered ... a sUberpace
ot XII; theretore llX induce. a Gau.sian ql1nder Dleuure
tollovs:
X in the norm
OD
(X'Ox)
as
X,
4
p{x c X:
{~(x),
• PX{x
where
~ , ••• ,~ c X.
extension 'to
(X,~)
Banach space
!
and
th~n
~
••• ,tn{x»
X:
(~{x),
€
D}
... ,tn(x»
n
R •
D is a Borel set in
p
and the norm
€
If
D},
(2.1)
p
haS a
.1s called ~ o-extenslon 2!. ~
II' II
is called admissible on
0-
sm. ~
X.
It is important to note that in the definition of an abstract Wiener
aeuure the meuurability ot the norm of the Banach space in question is
fundamental.
Indeed the relationship between the underlying Hilbert apace
and the measurable norm is an abstraction [4, p. 31] ot the relationship
between the Hilbert space ot all absolutely continuous functions in the
Wiener apace
in
~C[O,l]
C[O,l].
with square integrable derivative and the supnorm
Therefore the condition ot measurability of the norm in
the definition of an abstract Wiener measure can not be replaced by sClllle
other weaker condition like continUity.
(6},
Judging
from the introduction ot
,
Sato seems to imply that his result mentioned in Section 1 shoved that
t!!Very Gaussian measure on a real, separable or reflexive Banach space is
an abstract Wiener measure when restricted to a suitable SUbspace.
However,
in view of the fact that the measurability ot the norm in the definition
I
of an abstract Wiener measure is essential, his result simply shows that a
Gaussian meuure on a real, separable or reflexive Banach &dIlits a
restriction on a suitable subspace, which is a a-extension ot the canonical
Gaussian cllinder measure of a Hilbert space and the norm of the Banach space
is admi.sible.
We should also remark that although Sato's result appears
to be ..,er'/ signiticant in that it clears a new
war for the investigation ot
a Gaussian meuure in a Banach space, ret the strength of his result ls
sipificantly diminished due to the fact that the norm ot the Banach space
5
under consideration vas not shown measurable on the underlyinp; Hilbert
space.
Our result removes this deficiency and gives more insight in
studying a Gaussian measure on a Banach space.
1.foreover, our result allows
us to use important theorems about measurable norms obtained by Gross
[2, 3, 4] in studying a Gaussian measure on a Banach space.
That results
ot [2, 3. 4] vill yield useful results about Gaussian measures in BaDach
spaces 11 shOwn in Section
13 Gaussian measure
4.
!!!! abstract
Wiener measure.
In this section, we show that every Gausl1an measure
separable. or reflexive Banach space
separable closed subspace
i
X ada1ts a restriction
of X such that
\1i
\1
on a
\1i
on a
is an abstract Wiener
measure.
We will need two preliminary lemmas.
!:l!l
Lemma 3.1.
{ t j : J · 1, .•. ,n}
~ random variables
~
!!!!
~j'
J •
!!!..!. tinite familY 9! jointlY Gaussian
...
!il!!. ~ .!!!.2. defined .2!!. !!!!. probability space (0, s,p t.
1, ... ,t. 1 ~ k ~ n, ~ linearly independent ~ non-desenerate
i
( i · j:laiJ(j' i • k + 1 .... ,n, where
real numbers.
~
~ .t ~ detined!!:.2!!.
.a
t
a i " 's belons !2. the.!!i !
!2. ~ n-Eu21idean
Borel meuurable convex subset...!
C
~
a
t
t(lI) • «(l(w).· .. ,~(I1)',1;l~+l J(J(w),···."=lanjtJCw».
!2!:.!:Bl. !.. ("l .....an )
space
gL
.2!.
At
Then!2£.!:BI.
s)'Petric about ~ origip !:p!
.(0). !!. have
P{II: !n(II) C E} ~ P{w: !new) C !. + E}
where
!n(W). «(l(W) •••• ,tn(w».
Proot.
Without loss ot generality we can assume that
t , ••• ,tt are detined
l
k
k
on the canonical probability space CR ,8(n ) ,p) with t (xl .... ,~) • x '
J
J
k
.1-. 1, ••••t, where B(R ) is the clus ot Borel subsets ot the k-d1aenslonal
6
It
Euclidean ap&ee R.
In this setting the
~.p
k
~:
Rk
+
Rn becomes
k
'(~' ••• t~) • (~~ ••• t~'J;l~+l jXjt···.J;la~1xj). It ia clear that
;
18
linear and one to one.
sero and ; (Rlt.)
E ~
Since
If
is a sUbspace or Rn , it tollovs that , . E n t(Rk )
is also convex and symmetric about zero.
P{io ~ E} • P{~ ( ') • P{~
where ~. (tl' ••• '(k).
that .-l(l")
is convex, s)'1llllletric about
Bow
~-l(F)}
€
Since .-1: ~(Rk)
(3.2)
RIt i8 linear, it follows
+
is convex and symmetric about zero.
The
vector ~
satisties all the h;ypotheses ot Corollary 2 ot [1, p. 1721, therefore
we have
P{~ ~ .-l(F» ~ P{~
€
~ + ~-l(F)}
(3.3)
where :!!. ••-l(~). It i's easy to verlty' that ~-1(!.) + .-l(F) • ",,-l(A + l")
and '(Rk ) n (a + E) • a + tf.I(Rk ) n E • a + F. Using these tacts, (3.2)
•
and (3.3), we conclude inequality (3.1) as fo110vs:
-
-
-
P{~
€
E}
~ P{~ ~ .-1(~ +
• P{~
F)}
.!. + F}
€
• P{~ €
A+
E}.
It .ust be noted that inequality (3.1) Is trivially true when all
t
j
, j •
l •••• ,n. are degenerate at zero.
Therefore the conclusion of
Lesa 3.1 holds in both cases, nlllDe1y when all
degenate at zero or at least one ot the
tj
•8
t j ' j • l, ••• ,n, are
j.
1, ••••n, 1s non-
degenerate.
Lemma 3.2.
space
~ 1!. ~.!. Gaussian measure 2!!. A £!!! separable Banach
(x,II·ll x);
~!2!:. every
loI{X €
X:
E
> 0
Ilxllx
<
£l >
o.
(3.4)
T
Proot.
Since in a separable Banacb space 10be o-&1gebra generated by
"l1x,
norm open sets coincides vl10h
separabilU,. ot
equa1oion (3.4) makes sense.
X, we choose a sequence
{x : n • 1,2, ... }
n
Using
ot elements
ot X such 1oba1o
X·
where
•u
6(X ,(/2)
X
II ~
J
.1-1
6(X",U 2 ) . {x (X: fix -
J
r./2}.
Since
1 - p(X) < Y p{,6(X , £I2)}, it tollows 1oha1o there exists SOllIe positive
J
- .1-1
integer no such 1oha1o
p{t-(x
no
Using the separab11i10y ot
,c/2)}
>
o.
X once DIOre, we can tind a sequence
{tu: n - l,2, .•. } ot elements ot
X·
such that
IIx II x • SjP It j (x) I
tor every x
€
X.
Kow
p{x .: X: fix
• p{x
E:
~ p{x (
.
By
x: 8jPltj (x) I
x:
-limp{x.:
k-HD
<
Ilx
<
d
d
sJP It" (x) I ~ r./2}
x:
ItJ(x)I
< £/2,
-
.1 = l, ... ,k}.
(3.6)
LeIImaa 3.1. we have
p{x (X:
Itj(x) I ~
~ pIx. c X: It (x) -
.1
£/2, j • 1, ••• ,k}
t j (xn ) I ~ £/2, .1 • 1, ••• ,k} (3.7)
o
tor eTery positive integer k. From (3.5), (3.6) and (3.7). we obtain
e·
8
\l{x (X:
~ lim lJ{x
X:
E
k~
lI{x EX;
II:
Itj (x)
- t
(X
j
n0
SjPltJ(x) - tj(X
• ",{x «X: IIx. • lJ{6(x
Ilxll x < el
X
no
)
I~
)I
£/2, J
:II
1 .... ,k}
~ t!2}
no II x ~ £/2}
,£!2)} > O.
DO
The proof is complete.
Nov we are ready to state and prove the main result of this note.
As indicated in Section 1, the basic backgroWld of this
result 1s
developed by Sato in [6]; the only new thing here is to prove the
measurability of a certain norm.
For the sake of 'breVity, the notation
and terminology of LeDnas 1 through 5 of [61 will be used below without
turther explanation.
~
Tbeorem 3.1.
J!. 12.!..!. Gaussian measure £!t.!. real, separable·
reflexive Banach space
closed SUbspace
!
(X"
-
o-extension
2!. ~
!. II· IIX' :: II" IIx,/ x
i .!!!. 1h!. ~ II· I Ix·
Proof. Let
-
X and
respectively.
X
canonical Gaussian cylinder
1s measurable 2D.
! !!!! ! !!.
[6]
.
Then in view of Theorem 2 of [6] t the proof of our theorea
.
II· Iii
It is clear trom [6. p. 71] that
e.
l!i !!!. S
be the same as defined on pages 70 and 71 of
vill be complete if we can shov that
subset of
i
!.:.!t. there exists .!. sepa.rable Hilbert space ! .
measure ~ 2!.
dense in
exists !. separable
~ X ~.~ Hex). 1 !!!2:. I,!/X
abstract Wiener measure;
.!YS!l. ~ lAX' !!.!.
II· ! Ix). !!llm. there
g£.
X-, and moreover
X*
[4, p. 38] t the identity map on
itt
is measurable on
X.
can be identified with a
is dense in
X- •
X*
&8
regarded
By Corollary 1 of
densely defined JD&p of
, ...
9
x*
into random variables over the probability space
to a representative
a unique manner.
measure
D in
\.IX
P'
(X,Ui' J!i')
extends
of the canonical normal distribution over
X in
Furthermore. the con-esponding canonical Gaussian cylinder
~l, •••
satisfies (2.1) for any
,tn
X*, and any Borel set
€
n
R •
"..
Using leparabllity of the space
{~: n • 1,2, .•• } of elements of
X*
X, we can choose a sequence
such that
Ilx Iii· SjlP.I F;J (x) I
tor every x
X.
€
Since
X* ~ X*,
it tollows that the restriction
.3 :: t j I X belongs to X* for each J • Define the sequence ot pseudonorms
{II' II;:
Since
J • 1,2, •.. } on X as tollows:
+.1"
Ilx II; • 1+.1 (x) I·
tunc~ion f~ (x) = 1+.1 (x) I
c X*, the
[4. 1>. 3a] on X
tor each
is continuous tame function
1,2,... .' It tollows trom the detinition
j.
"..
ot
F
fJ
that the random variable
1s defined on the probability space
corresponding to
(X,UX'lJi)'
random variable • it tolloYs that for every
lJi{x
€
X:
Since
fj
It j I which
is
tj
is Gaussian
> 0
£
It.1 (x) I < d >
(3.8)
o.
Applying Theorem 1 01' [2. p. 406], COrollary 4. 5 of [3, p. 383] and (3.7),
,
Ilx 11.1
we have that
Ilx II
n
•
Ilx lin ~
is measurable pseudonorm for each
max Ilx 11.1'; then
1.s1~
.
•
Ilx III
Ilx II
is a pseudonorm on
n
I
+ ••• + Ilx lin·
il a measurable pseudonorm, it follo\ls that
hence
e"
sum
Since finite
j •
Let
X and
.
ot measurable pseudonorml
,
I!x III + ••• + Ilx lin
Ilx II
and
is a melLsurable !"seudonorm; moreover the random variable
n
·correspono.ing to Ilx II
is max I (x) I·
n
l~~n'
'-1
10
Since lim max
It" (x) I •
Ilxllx tor every x.:
n"'l~~
converges to ~he random variable
Ilxllx in probabili~y. Since
i
is
~ha~
> 0
Ui{X (
Thus we have a sequence
pseudonorms on
X:
{ II x II
X such ~hat
sequence of randCII variables
property
(x,UX'Ui)
Ui is Gaussian measure, it tolloys. by Lemma 3.2,
aeparable and
£
i~ tollova tha~
<llilln: n,- l,2, ••• } of random variables OD
the sequence
tor every
i,
Ilxlli > £) >
D
:
o.
n • 1,2, ••• } ot nondecreu1118 measurable
Ilxlli' the liJIU~ 1n probabilitY' ot the
{Ililln: n • 1,2, ... } exists and has ~he
tha~
Ui{x ~
The measurability ot
X:
Ilx IIi > d > O.
Ilxlli nov follovs from Coroll&r7 4.4 ot (3, p. 383).
14. !!!!. support 2! A Gaussian measure.
In
~hls sec~loD,
ve obtain the support ot a Gaussian aeasure defined
on a real,. separable or reflexive Banach space.
~bat
it U 11 a Gaussian measure on a real, separable or retlexive Banach
space X
vhere
Sepcltically, we shoY
-
~hen ~be
support of
X is the same
&8
-
U 1s the separable closed sUbspace X,
in Theorem 3.1.
Hilbert space, ve show that the
suppor~
Specializing this result to a real
ot a Gaussian measure U on a
real Hilbert space is the closed subspace spanned by the eigenvectors
c~rrespoDding to
the nonzero eigenvalues ot
ot u. This result vas first obtained
ditterent
_~hod.
Gaueslan measure
aD
~he
by I~o
covariance operator Su
[5] by a tr1cq and
en~1re~
We also give a necessary and sufticient condition tor a
U on a real, separable or retlex!va Banach space to be
abstract Wiener measure.
.'.
,"
.
11
.&!!. J!.
Theorem 4.1.
~.!. Gaussitm measu;e gn,.!. real, separable .2!:.
!. !h!.!!. ~
reflexive' Banach space
closed subsp!ce
Proof:
i
2!. .1,
!
where
We already know that
i
support
or
J!.
.!!.!h!. separable,
.!!. ~ !!!!!!: !.!. in. Theorem ~
is a closed SUbspace ot
i
J,li(X).
with
The proof will be complete, if we can show that every ~pen subset ot
But this follows bY' Corollary 4 ot [4tP. 40].
has positive measure.
1.
i
Hence
the proot 1s complete.
~ J!. ~!. Gaussian measure 2!l.!. real, reflexive 2!:
Theorem 4.2.
!. !h!!'!..!. neeossm
,separable Banach !'Dace
J!.
~
sUfticient condition
E!.!!!. abstract Wiener measure !!. ~ tor every
J (2(X)d~(X)
•
X
0 implie~ Sex)
t
It tor every
Proof.
f
X
(x)
= 0,
(2(x)~ (x) • 0 implies
then, it follows f'rom [6, p. 70], that
same as in Theorem 3.1.
Therefore
Conversely, suppose
to prove that tor every
}.l
is
X· !!!!. condition
= o.
X* the condition
€
~
t
~
1J
Ron
X.
X is
X where
the
18 'an abstract Wiener measure.
abstract Wiener measure on
X, we have
t (X* the condition J t 2 (x)c1).I(X) • 0 implies
X
(x)
=O.
From Corollary 4 ot [4, p. 40], i1# follovs that
IUpport ot
ll.
Suppose there exists some
t (X.
X is the
J (2(X)d\.l(X)
such that
• 0
X
but (x). 0; then,
proper subsPace ot
by
X.
Theorem 3.1, -the support ot
This is a contradiction.
II
i
is
whicb 1s a
Therefore ve bave that it
J (2(X)dll(X) • 0 then t(x) = O.
X
The following corollary is an immediate consequence 01' the above
( c X*
and
theorem.
Corollary 4.1.
-Let
&.I
-.
reflexive Banach space
!2t evety ~ (X· ~
be 8. Ga.ussian measure on a real
!. TI:!!m. ! !!. ~ support 2!.
--~
condition
f
X
2
t (x)dJ,J(X) • 0
separable, or
.....
J!.
!!. ~ only !!
1!p11ss
~
= o.
12
Theorem 4.3.
J£.l .w.
(Ito)
l'l!!m. ~ support 2! ~
(X, <. >).
.!I!!:!l ~ ~ eisenvectors
eisenva.lues
Proot.
~.!. Ga.ussian measure oil!. ~ Hilbert space
2!.:!U!!.
sP
is
{Zn }, ~ closure g! ~ 1!near
{Z : n • 1,2, ••• }
correspondins
n
2!.
~
covariance operator
~ non~ero
jL.
X is a Hilbert space, it follows, frOlll Theorem 4.1, that
Since
the support ot
i,
1s
\J
i
where
is the seme
in Theorea 3.1.
8.8
sP
in order to complete the proof ot the theorem, ve must shOY
Let
{t: a
CI
E
ot [6, p. 66]; and h
ot [6].
~
h}
be the maximal subset ot
o
and
a
t a(.) •
Further let
z ,za>. Xa
It (x)ta(X)du(x),
1&a
<z ,. >,
Z
a
a e: X
Z ,ZD>·
101 CI
IiJ
Ita(x) F;a(x)du(x)
n. 1,2.... •
\In m
&nm
is Kronecher's
f
l
X"
hn} •
-
given
z
n
X
o
CI
-
vhere
+8,
a, B €
~: X ... X
A.
by
x·
a
-
z (X.
n
{x EX:
e: X, ve BlUst show that
= {tCI :
CI t:
h.
Since
}
(4.1)
~ (.) • <z n' • >
A }
0
First ve prove that
X.
it 1s enouch to show that each
i . ClCI\-1\
_f' a
t:
6; and it is easy to verity that
~
-sp
a
68 and 10
(X)E (x)d,,(x) • ~ 6 ,
"
nnm
{E : n • 1,2 •••• }
Bov ve shall shOY
X.
Then we ha.ve
<S z .z > •
vbere
tor evert
1
• 0, ~
X
Define the continuous linear maps
tor each
{zn} •
condition 3.1 of (6] becomes
IltaIIX*· IIzallx.
<S
Thus
xtt which satisfies (3.1)
be the aeme as defined on pages
X
' ..
t
CI
this 1a not true, then there exists
b n} ~
X,
clearly
Since. by Lemma 2 ot [6],
(x) • OJ; in order to ahOY that a
f,; (z ) • 0
CI
8P
n
80llle
tor every
Cl
E
O
A-A
a e: A-A •
o
o such that
Suppoae
t a (z)
o n
+O.
13
It follows that
<z,z
n Q
o
>'
o~ ~t, by (4.1) and (4.2), <5 z,z > . 0
~ n
Q
o
A <z;z >. O. This contradicts the fact that both A
n n a n
or
o
-
<z , z > are nonzero. Therefore z (X
n a n
order to prove
-
there exists a
7
o
-
71 ,1 2 ( X, Y'1
X.:. sp h
f:
(X
-
liP hnl
orthogonal subspace to
'l'berefore
Ilzll x
a
1
S z • 0
~
l~her
such that
and 1'2
4 sp h n )'
y
i t follows that
z
E:
-
X.
Q
1
J.
z.
Let
n
<8 z,z > • 0
)
n
c
-
X.
In
( A-A.
f
7· Y + Y ,
2
l
-
Bp hnl
Y2
-n::-TT', then
1112 1I
for every
n
~
<Sz,z>.
o
Bp h
Write
sp {zn} , where
£
sp {z }.
and so
since
--
and hence
}, we assume the contrary. 1.e., we suppose that
{~a:
• 1, by the definition of the set
(A.
that
n
and
Q £
a
z
£
J.
£
II.
Al, z ·
But then
Therefore we must have
t 11 Cz)
-
= 1,
-
J.
sp {z 1 •
n
Since
za
t 2Cx)d~(x) • 0, where t
X a1
ill the
for some
l
al
C·)· <z,·>,
which contracts the tact
1
X ~ sp {z }.
n
The proof is complete.
"
14
REFERENCES
[1]
[2]
T. W. Anderson, The intep::!"a.l of a symmetric unirlOda1 f'unt':t10n over a
symmetric convex set and some probability inequalities, Proc. Amer.
Math. Soc., 6 (1955), pp. 170-176.
L. Gross, Integeration and nonlinear transformations in Hilbert space,
TranI. Amer. Math. Soc., 94 (1960), pp. 404-440.
[3]
L. Gross, Measurable functions on Hilbert space, '!'ranI. Amer. Math.
Soc., 105 (1962), pp. 372-390.
[4] L. Gross, Abstract Wiener spaces, Proceedings of the Fifth Berke1e:r
Symposium, University of California Press, Berkeley, California,
Vol. 2, part 1, (1965), pp. 31-42.
[5) K. Ito, The topological support of Gauss measure on Hilbert space,
Nagoya Math. J., 38 (1970), pp. 181-183.
(6]
H. sato, Gaussian measure on a Banach space and abstract Wiener
measure, Nagoya Math. J., 36 (1969), pp. 65-81.
Departments of Mathematics and Statistics,
University of North Carolina, Chapel Hill, H. C.
....