2042 CHAPTER 61. PROBABILITY IN INFINITE DIMENSIONS

This has shown that

P({

ω ∈Ω :∣∣∣∣Sn (ω)−Spm (ω)

∣∣∣∣> 2−m})< 2−m (61.9.47)

for all n > pm. In particular, the above is true if n = pn for n > m.If{

Spn (ω)}

fails to converge, then ω must be contained in the set,

A≡ ∩∞m=1∪∞

n=m{

ω ∈Ω :∣∣∣∣Spn (ω)−Spm (ω)

∣∣∣∣> 2−m} (61.9.48)

because if ω is in the complement of this set,

∪∞m=1∩∞

n=m{

ω ∈Ω :∣∣∣∣Spn (ω)−Spm (ω)

∣∣∣∣≤ 2−m} ,it follows

{Spn (ω)

}∞

n=1 is a Cauchy sequence and so it must converge. However, the set in61.9.48 is a set of measure 0 because of 61.9.47 and the observation that for all m,

P(A) ≤∞

∑n=m

P({

ω ∈Ω :∣∣∣∣Spn (ω)−Spm (ω)

∣∣∣∣> 2−m})≤

∞

∑n=m

12m

Thus the subsequence{

Spn

}of the sequence of partial sums of the above series does

converge pointwise in B and so the dominated convergence theorem also verifies that thecomputations involving the characteristic function in 61.9.45 are correct.

The random variable obtained as the limit of the partial sums,{

Spn (ω)}

describedabove is strongly measurable because each Spn (ω) is strongly measurable due to each ofthese being weakly measurable and separably valued. Thus the measure given as the lawof S defined as

S (ω)≡ limn→∞

Spn (ω)

is defined on the Borel sets of B.This proves the theorem.Also, there is an important observation from the proof which I will state as the following

corollary.

Corollary 61.9.15 Let (i,H,B) be an abstract Wiener space. Then there exists a Gaussianmeasure on the Borel sets of B. This Gaussian measure equals L (S) where S (ω) is thea.e. limit of a subsequence of the sequence of partial sums,

Spn (ω)≡pn

∑k=1

ξ k (ω)ek

for {ξ k} a sequence of independent random variables which are normal with mean 0 andvariance 1 which are defined on a probability space, (Ω,F ,P). Furthermore, for anyk > pn,

P({

ω ∈Ω :∣∣∣∣Sk (ω)−Spn (ω)

∣∣∣∣> 2−n})< 2−n.