A NNALES DE L’I. H. P., SECTION B
C HRISTER B ORELL
A note on Gauss measures which agree on small balls
Annales de l’I. H. P., section B, tome 13, no 3 (1977), p. 231-238
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Ann. Inst. Henri
Section B :
Poincaré,
Vol. XIII, n° 3, 1977, p. 231-238.
Calcul des Probabilités
et
Statistique.
A note on Gauss measures
which agree on small balls
Christer BORELL
1. INTRODUCTION
compact metric space K and
singular Radon probaall balls ([5], Th. II, [4] ). Therefore,
since K is isometric to a compact subset of the Banach space C(K), we can
find two singular Radon probability measures and v on C(K) satisfying
the condition
There exist
bility
(Co)
a
measures on
for
K which agree
a E
=
Here
p.
on
C(K) there exists
,u(B(a ; r)) v(B(a ; r)),
every
two
a
b
0
>
0 such that
r
~ .
B(a ; r) denotes the closed ball of centre a and radius r. (Compare [6],
326, and [9].)
The main result of this note shows that two Gaussian Radon measures
C(K) (or any Banach space) coincide whenever the condition (Co)
holds (Theorem 3.1). Moreover, we prove that two Gaussian Radon
measures on a Banach space are equal, if they agree on all balls of radius
one (Theorem 3.2). The same theorem also gives a positive result for dual
Banach spaces, equipped with the weak* topology.
Finally, I am grateful to J. Neveu, H. Sato and F. Topsøe for a very
stimulating exchange of ideas about the group of problems considered
in this note.
on
.
2. THE REPRODUCING KERNEL HILBERT SPACE
OF A GAUSSIAN RADON MEASURE
In this section it will always be assumed that E is a fixed locally convex
Hausdorff vector space over f~. The class of all (centred) Gaussian Radon
Annales de l’Institut Henri Poincaré - Section B - Vol. XIII, n° 3 - 1977.
232
C. BORELL
by ~(E) (~o(E)). In the following, all non-trivial
proved or, otherwise, they can be found in e. g. [3].
Let p E ~(E) be fixed and denote by b the barycentre of p. Set
the closure of E’ in
respectively.
,u( . + b) and
for
Then
E E. The
every ~ E E2(,u), the measure ~ 0 has a barycentre
For brevity
map A : E2(~u) -~ E is injective. Its range is denoted by
we write A-1 h
h, h E H(,u). Obviously, the scalar product
measures on
E is denoted
statements will either be
=
=
=
makes
into a Hilbert space, the so-called reproducing kernel Hilbert
of
The
closed unit ball
of
is a compact subset of E.
,u.
space
Moreover,
...
Observing that
we
have the
following
useful
v if O(,u)
Then ~u
O(v).
from measures of
now on will be to determine
sufficiently many « balls ». A weak result in this direction follows from
e. g. [3], Th. 10. l. Theorems 2. 2 and 2. 3 below yield stronger conclusions.
Before proceeding, let us introduce
THEOREM 2 .1.
Our strategy from
v E
~o(E).
Moreover, in the following, measurable always
THEOREM 2 . 2.
convex,
~
Let ,u
and measurable subset
of
=
=
means
and suppose V is
E such that ,u(rV)
a
>
Borel measurable.
bounded, symmetric,
0, r > 0. Then
for small r > 0 is known.
In many special cases, the behaviour of
A. Shepp [8] give some
L.
and
For example, J. Hoffmann-Jørgensen [7]
very precise estimates when E is a Hilbert space and V the unit ball of E.
To prove Theorem 2.2,
we
need two lemmas.
Annales de 1’Institut Henri Poincaré - Section B
233
A NOTE ON GAUSS MEASURES WHICH AGREE ON SMALL BALLS
LEMMA 2.1.
~
E
[3],
Cor. 2.1
(Cameron-Martin’s formula).
- [2],
Cor. 2.1, Th. 6 .1. For any ,u E ~o(E)
all measurable subsets A and B
In
of
E.
particular,
whenever A is symmetric, convex, and measurable subset
PROOF
OF THEOREM
2 . 2.
Cameron-Martin formula,
-
we
Let
us
first
assume
that
of
a E
have
Moreover, the Jensen inequality yields
Since
it follows that
We
For any
~o(E)
LEMMA 2 . 2.
for
-
now
prove the estimate
To this end
let ~ E E’ be fixed and set h
=
A~. Then (2. 2) gives
Moreover, the Cameron-Martin formula yields
Vol. XIII, n° 3 - 1977.
E.
By the
234
C. BORELL
By applying Lemma 2.2,
we
have
and hence
By choosing ~ E E’ close to a in E2(,u), the estimate
This proves (2 .1 ) when a E
Let now a E supp (,u)BH(,u). Then, for every n E
such that
(2. 3) follows
there exists
at once.
a
~n E E’
and
respectively. Set an
and note that
=
By applying Lemma 2.2,
we
have
Furthermore, observing that ,u is symmetric, the Cameron-Martin formula
yields
Using (2.4),
it follows that
Here, by the first part of the proof, the left-hand side equals exp
Clearly,
this
(-
expression
converges to zero as n tends to plus infinity. This
when
a
E
proves (2.1)
supp
Finally, the case a E EBsupp (p) is
trivial. This completes the proof of Theorem 2.2.
Annales de 1’Institut Henri Poincaré - Section B
235
A NOTE ON GAUSS MEASURES WHICH AGREE ON SMALL BALLS
We also have
THEOREM 2. 3. Let ,u E ~(E) and suppose V is
subset of E with positive -measure. Then
Proof
Suppose
a
bounded measurable
- Without loss of generality it can be assumed that J1 E ~o(E).
As in the proof of Theorem 2 . 2, we have
first that a E
and
respectively.
We
now
Hence
prove the estimate
-
B
To this end let
it follows that
Using
A~. Then, assuming t
>
0,
(2.6) yields
By choosing ~
a E H(,u).
Vol.
=
the trivial estimate
the relation
Let
~ E E’ be arbitrary and set h
/
now a E
to
close to a in
EBH(,u). Then,
XIII, n° 3 - 1977.
E2(,u),
we
for every
get (2.7). This proves (2. 5) when
nE
there exists
a
~n E E’
so
that
236
and
C. BORELL
çn(a)
=
1, respectively. Since
it follows that
By letting n tend to plus infinity, we get (2. 5) for a E
proof of Theorem 2. 3.
This concludes
the
3. APPLICATIONS
The results
proved
in Section 2
apply
to any
locally
Hausdorff
convex
vector space. In order to be concrete, however, we here restrict ourselves
to Banach spaces and dual Banach spaces equipped with the weak* topo-
logy respectively.
THEOREM 3 .1. - Let E be a Banach space and suppose ,u E
Moreover, assume there exists a function ~ : B(0 ; 1) ~
such that
Then ,u
=
Proof
+
oo[
v.
- Let
c
denote the
Lemma 2 . 2. Hence - cesupp vo
k E B(c ; 1) n H(v), we get
vo(B(0 ; r))
~
vo(B( -
c;
of
barycentre
by
Since
and
]0,
r))
>
0,
=
v
and note that
H(v) [3],
r >
Cor. 8 . 2.
By choosing
0, the relation
be true for all 0
min (5(c - k), b(O)). By letting r tend to zero
r
from the right and using Theorem 2.2, we get - ceH(v). Moreover,
must
for every 0
r
min (~(a), (5(0))
Theorem 2.2 therefore yields that
1. Another
HM
=
application
of
H(v) and
Annales de l’Institut Henri Poincaré - Section B
237
A NOTE ON GAUSS MEASURES WHICH AGREE ON SMALL BALLS
Now
and letting t tend to zero, we have
Theorem 2.1 therefore implies that ,u
choosing a
=
=
tc
c
=
=
v.
0. Moreover,
This proves
Theorem 3.1.
a Banach space or a dual Banach space
let ,~ E ~o(E)B ~ Dirac measure
with
the
weak*
Moreover,
topology.
equipped
at 0 ~ and v E ~ (E) be such that
THEOREM 3. 2. Let E either be
and
v.
1+’here K > 0 is a fixed constant. Then ,u
The condition (3.1) is, of course, automatically fulfilled, if E is a Banach
space. Note also that the closed unit ball B(0 ; 1) is weak * measurable
when E is a dual Banach space.
=
Proof - Theorems 2.3 and 2.1 tell us that ,u vo. Let c denote the
barycentre of v. It only remains to be proved that c 0. Suppose to the
contrary that c ~ 0. Let first a E EB { 0 } be arbitrary and choose p = pa E N +
=
=
+ 1. Then
such that
For every n E
with n
>
K,
we
therefore get the
following chain of equa-
lities
1))
(3 . 2)
=
vo(B(npa ; 1)) v(B(npa
p(B(npa + c ; 1))
=
=
By assuming that
that
c E
again using
In the next step,
Theorem 2.3
and
we
+ c ;
_
,u(B(n(pa
+
c) ; 1)) .
Theorem 2 . 3, we deduce
and p
p~ in (3 . 2) and get,
applying
set a
=
c
Hence c
0, which is a contradiction. This,
the proof of Theorem 3.2.
concludes
and
=
1))
... -
=
finally,
shows
that ,u
=
v
REMARK 3 .1. - Theorem 3 .1 is true for a dual Banach space E, equipped
with the weak * topology, if we assume that
r)) > 0, r > 0. However,
under these conditions both ,u and v extend to Gaussian Radon measures
the Banach space E [1], Th. VI, 2 ; 1. The result is thus already contained
in Theorem 3. l.
on
Vol. XIII, n° 3 - 1977.
238
C. BORELL
REFERENCES
[2]
[3]
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C. BORELL, Gaussian Radon measures on locally convex spaces. Math. Scand., t. 38,
[4]
R. B.
[5]
p. 224-225.
R. O. DAVIES, Measures not
[6]
J.
[1]
1976, p. 265-284.
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on
balls
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[9]
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1976.
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(Manuscrit reçu le 4 mai 1977)
Annales de 1’Institut Henri Poincaré - Section B