MEASURES ON C( Y) WHEN Y IS A
COMPACT METRIC SPACE
JAMES KUELBS1
1. Introduction.
Let Y be a compact metric space and C( Y) the
space of real-valued
continuous
functions on Y with the uniform
topology. The minimal sigma-algebra
containing the closed subsets
of C(Y) will be called the Borel subsets of C(Y). If F(p(l, ■ ■ ■, k)),
piE Y, (where p(l, ■ ■ ■ , ft) stands for pi, • • • , pk) is a consistent
family of finite-dimensional
Gaussian distributions
then sufficient
conditions are given to assure the existence of a measure p on the
Borel subsets of C( Y) whose finite-dimensional
distributions
are those
postulated.
It will also be shown that such a family of distributions
always exists and that Levy's Brownian motion with parameter
from
the Hilbert space h [l, pp. 293-298] can be taken to have continuous
sample paths when the parameter
is restricted to certain compact
subsets of 1%.
2. The product
space X=
YLt-i [0, 1/2*] is assumed
topology induced by the usual l2 norm ||-||. Let F(p(l,
to have the
■ ■ • , ft)),
pi E Y, be a consistent family of Gaussian distributions.
If for some
homeomorphism
0 of F into X there exists another consistent family
of Gaussian distributions
G(q(l, • • • , ft)), qtEX, with the property
that F(p(\,
• • • , k))=G(<p(p(\,
■ ■ ■ , ft))) (where <p(p(l, • • • , ft))
stands lor<p(pi), • • • , 4>(pk) and
f f | s2-s1\2dG(q(l,2))(s1,s2) ^C\\q2-qi\\°
where
C>0,
family F(p(\,
a>0
are independent
of q\, q2 in X, then
we say the
■ ■ • , ft)), piEY, can be extended to X.
Theorem
I. If Y is a compact metric space and F(p(l, ■ ■ ■, ft)),
PiE Y, is a consistent family of Gaussian distributions which can be
extended to X, then there exists a measure p on the Borel subsets of C( Y)
whose finite-dimensional
distributions are those given.
Corollary.
// F= YLt-i [ak, a*+ 1/2*] where {ak} is in l2 and
Received by the editors November 3, 1965.
1 This research was supported by a National Science Cooperative Fellowship at
the University of Minnesota and is a portion of the author's doctoral dissertation
under Professor R. H. Cameron. Additional work on it was done at the University of
Wisconsin.
248
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MEASURESON C(Y) WHEN Y IS A COMPACT METRIC SPACE
249
F(p(\, ■ ■ -, k)),pi(EY, is the family of Gaussian distributions with mean
vector zero and covariance matrix
V=(pi,)
where 2z\7 = ||£;||+||/>y||
— \\pi —p,\\ for i, j= 1, • • • , k, [l, p. 293], then there exists a measure
p. on C(Y) whose finite-dimensional
distributions agree with those given.
Proof.
Let <p((yi, y2, • ■ • )) = (y\ —ai, y2 —a2, ■ ■ ■ ). Then <p is a
homeomorphism of Y onto X. Let G(q(l, ■ • • , k)) = F(p(\,
where pi=<p~l(qi) for i = l, ■ ■ ■, k. Then
• ■ ■, k))
f f \s,-s1\2dG(q(l,2))(shs,)
= J J \h-h\2dF(p(l,2))(h,h)=11^1-^11
and the family F(p(l,
■ • • , k)), £,£ Y, can be extended to X with
respect to <f>.Hence the corollary follows from the previous theorem.
As a result of this corollary we see that Levy's Brownian motion
with parameter from the Hilbert space 4 can be defined to have continuous sample paths on subsets of the form £l£L, [ak, ak-\-l/2k].
In fact, the conclusions hold if Y is any compact subset of such a set.
However, it is known [l, p. 293] that sample function continuity does
not hold for arbitrary subsets of h-
3. The proof of our theorem will be obtained from the following
sequence of lemmas. Throughout
the discussion it is assumed that
X is as defined above and that C(X) is the space of real-valued continuous functions on X with the uniform topology. Since X is compact C(X) is a complete separable metric space and the results of
Prokhorov [2] can be applied.
The Borel sets S of C(X) will be the minimal sigma-algebra
containing the closed sets J? of C(X). The space of all probability measures on the measurable space (C(X), S) will be denoted by VH(C(X)).
If {mk\ is a sequence of elements in 3K(C(X)) we say {mk} converges
weakly to m in 3H(C(A')) provided that
lim f
n-.« J ax)
for every bounded
F(f)dmn= f
continuous
J c(x>
functional
F(f)dm
F on C(X). A metric is de-
fined on 3\l(C(X)) in the following way. If mi and m2 are in SflTwe let
L(mi, w2)=max{5i,
S2} where
5i = inf {e > 0: mi(F) < m^F') + e for all FGf),
S2 = inf {e > 0: m2(F) g mi(F') + e for all FGJJ,
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and
250
JAMES KUELBS
F' = \f G C(Y):
[April
max | /(*) - g(x) | < e for some g G FJ.
In [2, pp. 168-170] it is shown that i-convergence
and weak convergence are equivalent in 31l(C(Y)) and that 9H(C(Y)) is a complete
separable metric space in these topologies. If 0 is a continuous
mapping of C(X) into C(Y) and wE3TZ(C(A")) we will denote by
m* the element of 9TC(C(Y))such that »n*(4) = m(6~l(A)) for all 4 £5.
If 5>0and m(H)>l-e,
where
J? = \fEC(X):
then i(w,
m9)^jmax(e,
max |/(«) - 6(f)(x) |< 8L
5). This estimate
is essential
to us; its proof
is in [2, p. 167].
Lemma 1. Let H=YLj=i[a3>
bj] where — » <aj<bj<
00 /or j
= 1, • • • , N and suppose f(xi, ■ ■ ■ , xn) is a real-valued function on H.
Then there exists a continuous function f on H such that /=/
ow
(P= {(*ii • • ■ > *w): *y=«y or */ = &/} awd maxpeH/(p)
=maXj,e(p/(^)
= minpe(p/(p) =minper/ /(/>). In fact, we take / /o oe as follows:
?(*i,• • •, *x) = X)/(yi. ■■•, yw)
i-l
(xi -
aQ'tyi
~ si)1-2'
• • • (agjy — aif)'*(bif
-
xN)l~^
(bi — ai) • • • (bN — aN)
where {(y\, ■ ■ ■, yh): «= 1, • • • , 2W}=(P, s}= 0 if yj = a,-, sj = l if
y'j = bj, and we assume (Xj —aj)°=(bj
Proof.
—Xj)° = l.
It is clear that / is continuous
on H and that /=/
on (P.
Theprooithatmaxp(=ii?(p)=maxpE(pf(p)=m'mpeQ>f(p)=minpeH}(p)
follows by induction on N.
Let
(Pjv={(xi,
• ■ ■ , xat, 0, • • • ):
xk=j/2N,
0^jS2tf-h}
for
N= 1, 2, • • ■. The number of points of X which 0"Vcontains will be
denoted
by ord((Piv). Hence we have ord((PN)^2N(N+l)l2.
N
N
.
.
Let
N
Sn = II [0, 1/2 ], and HN=H [al, a'k+ 1/2 ]
*-i
*=i
where a{=j/2N and j is an integer depending on i
0^/^2JV-*-l
for k = l, ■ ■ • , N. Then there are
/zlr and by relabeling the iJJv, if necessary, we have
of the Hjf for 1 ^ig2ww-1)'2.
Moreover, the Hjf
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and ft such that
2JV<JV-1)'2
distinct
HN as the union
overlap only on
1967I
MEASURESON C{Y) WHEN Y IS A COMPACTMETRIC SPACE
their boundaries
any real-valued
251
when they are considered as subsets of Rn- If / is
function on X we construct
the function fN on X
such that ftr(xi, • • • , Xn, xn+i , ■ ■ ■) =/•(*,•, ■ ■ ■, xN, 0, 0, • • • )
for (xi, ■ ■ ■ , Xn, 0, • • • ) in HNX(0,
0, • • • ) and where
}i(xi, ■ ■ ■ , xn, 0, • • • ) is defined on HNX(0, 0, • • • ) by using
Lemma 1 on/(xi, • • • ,Xn, 0, • • • ) and the obvious homeomorphism.
Then /# is continuous on X and fN =f on 0~V. Moreover, the maximum and minimum of /# over X are both attained on 6~V-
Definition.
If / is in C(X) let GN(f) —jN where fN is defined as
above. Let SN = GN(C(X)) for N=l, 2, ■ ■ ■.
From the above considerations it is quite clear that Gn is a continuous
mapping of C(X) onto the closed linear subspace Sn of C(X) and that
Sn is homeomorphic
to Rl where L = otc1(<Pn). Let G(q(l, ■ ■ ■ , k)),
qiE:X, be a system of consistent Gaussian distributions.
If E is any
Borel subset of RL and /= {/£Sjv: \j(qi), • • • , /(?z)]GP}
where
{si, • • ■ , <Zl} = (?n then we define mN(I) = fs dG(q(l, ■ ■ ■, L)).
Then m^ is a measure on the Borel subsets of Sn and since Sn is a
closed subset of C(X) we can extend mN to a measure on the Borel
subsets of C(X) simply by letting mN(C(X) — Sn) = 0. That is, my
so extended is an element of 9TC(C(X)). Moreover,
mN = mN-i for N = 2, 3, • • • .
if 8 = GN~i then
Lemma 2. The sequence {mN} is a Cauchy sequence in 911(C(X))
provided
ff
\s2-s1\2dG(q(l,2))(susi)
fS C||?1 - o2||«
for all gi, q2 in X where 00
and «>0 are constants independent
qx, q2 and \\qi —?2|| is the usual h distance between qi, q%.
of
Proof. We will examine L(mN, mN-i) by noting that L(mN, mN-i)
= L(mN, m\) where 8 = Gn-i- Since mN(C(X) —Sn) =0 we have
J = mN\fEC(X):
max | 0(f)(q) - f(q) | > 1/A2(
(
= mN\feSN:
(
aex
\
max | 0(f)(q) - f(q) \ > 1/N2).
aex
\
Now if f(ESN then 8(f) is in SN-i and
max \f(q) - 0(f) (q) | = max f max \f(q) - 0(f) (q) n
where Z = 2(JV-1)(Ar-2)'2
and Zv = HkiX[0,
more, if f(E.SN Lemma 1 implies that
1/2^] X(0, • • • ). Further-
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252
JAMES KUELBS
max |/(g)-<?(/)(?)
| g |/(gi)-/(<Z2)|
qeZi
where qiE&N^Zi,
[April
q2E<?N-i<^Zi, and qi9^q2 depend
on /. Hence
K
J =gEwtfj/G-S*:
i-l
max \ f(q) - 6(f)(q) | > 1/A2/
/
«6Zi
\
and we see that
mN\fESN:
I
max \f(q)-8(f)(q)\
> \/N2\
qezf
\
=
Z
^{/6^:
| /(<?,)- f(q2) | > 1/A2}
where A = {(glt g2): giG^HZ,-,
^G^at-i^Z,-,
q\^q2\. Now (gi, g2)
G^4 implies ||gi —^H <Nxl2/2N~l so for (gi, g2)G-<4 and 7 a positive
integer we have
mN\fESN: |/(?0-/(?0|
>VA2}
= »**{/£ Sw: |/(gi) -/(g2)
I2"'* > (1/A2)2^}
^NiNy C \f{ai)-f(q2)\2^dmN
= Y^[Var(/(0i)
-/(g2))]^[l-3-5
= CN*Ny[N1i2/2N-i]Ny[l-3-5
Letting
g3*-2"
ord(^4) denote
and K-ord(A)
■ ■ ■ (2Ny - 1)].
the number of elements
=2N*i2+3Ni2+1. Hence
/ = C-2N2'2+3Ifi2+l-NiN"'[l-3-5
■ ■ ■ (2Ny
• • • (2AT - 1)]
of A we have ord(^4)
-
l)][N1i2/2N-1]Ni"
and choosing 7 so that 7-a2:1
we see that i^expj
—A7} for N
sufficiently large. As a result of the estimate
[2, p. 167] we have
L(mN, mN-\) ^maxjr"1',
N~2} for N sufficiently large and hence
{mN} is a Cauchy sequence in the i-metric so our proof is complete.
If G(q(\, • • • , ft)), qiEX, is a consistent family of Gaussian distributions satisfying the conditions of Lemma 2 then jjnj} is a Cauchy
sequence in 3H(C(X)) and there exists an element m in 3TlC(X) such
that {mff} converges weakly to m. Moreover, mffaM will converge
weakly to mGM and since m^GM = mM for N= M we have mGM = mu-
That is, if \q\, • ■ ■ , qk} Q(Pn for some N then
m{fEC(X): \f(qi), ■■■,f(qk)}E E) = f dG(q(l,■■■, ft))
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1967]
MEASURES ON C{Y) WHEN Y IS A COMPACT METRIC SPACE
253
for any Borel subset E of Rk. To show that all the finite-dimensional
distributions
of m coincide with those postulated
follows quite easily.
The following theorem summarizes
the results of this section.
Theorem
2. If G(q(\, ■ ■ ■ , k)), qiE:X, is a consistent family of
Gaussian distributions then there exists a measure m on the Borel subsets
of C(X) whose finite-dimensional
distributions
coincide with those
postulated provided that for all q\, q2 in X
f f | 52- si\2dG(q(l,2))(si,Si) S C\\qi- ?2||«
where C > 0 and a > 0 are independent
4. Proof of Theorem
of q\, q2.
1. Since the family
F(p(l, ■ ■ ■, k)),
/>.G Y, can be extended to X there exists a consistent family of Gaussian distributions
G(q(l, ■ ■ ■ , k)), q^X,
which satisfies the condi-
tions
of Theorem
2 and
F(p(\, ■ ■ ■ , k)) =G(<j>(p(\, ■ ■ ■ , k)))
where <pis a homeomorphism
of Y into X. Let m be the measure on
C(X) obtained as a result of Theorem 2. Since <pis a homeomorphism
of F into X we have 0(f) = /(<£(')) mapping C(X) continuously
onto
C(Y). Then m° is a measure on the Borel subsets of C(Y) and if
/= {gGC(Y):
[g(pi), • • • , g(pk)]<=E} where pi, ■ ■ ■ , pk are in
Y and £ is a Borel subset of Rk we have
m?(I) = m(0~l(I))
= m{fG C(X): [0(f)(pi), • • • , 0(f)(pk))G E)
= f dG(cb(p(l,
• • •, A)))= f dF(p(\,■■■, k)).
Hence p = me has the finite-dimensional
the theorem is proved.
5. Since Y is a compact
distributions
postulated
and
metric space there exists a homeomorphism
cj>of Y into X. Consequently, if G(q(l, ■ ■ ■, k)), g.GA", is the family
of distributions
mentioned
in the proof of the Corollary
to Theorem
1
then the family F(p(l, ■ ■■, k))=G(cb(p(l, ■ ■ ■, k))), p^Y,
satisfies Theorem
1 so the existence
question
mentioned
in the intro-
duction is easily handled.
As a final remark we mention that the conclusions
hold if the hypotheses are such that the integral
f f \s2-s1\2dG(q(l,2))(si,s2)
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of Theorem
2
254
JAMES KUELBS
is less than or equal to C\ qi —q2 " where OO, a>0 are independent
of qi, q2 and \qi —q2\ = ]Ci-i °<~a'\
when qi*=(ai, a2, ■ ■ ■ ), q2
= (bi,b2, • • •).
Bibliography
1. P. Levy, Processus stochastiques et mouvement brownien, Gauthier-Villars,
Paris
1948, 275-298.
2. Yu. V. Prokhorov,
Convergence of random processes and limit theorems in prob-
ability theory, Teor. Verojatnost. i Primenen. 1 (1956), 177-238 =Theor. Probability
Appl. 1 (1956), 157-214.
3. I. J. Shoenberg,
On certain metric spaces arising from euclidean spaces by a
change of metric and their imbedding in Hilbert space, Ann. of Math. 38 (1937), 787-793.
University
of Wisconsin
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