61.8. GAUSSIAN MEASURES FOR A SEPARABLE HILBERT SPACE 2021

Then for the index set equal to N the measures satisfy the necessary consistency conditionfor the Kolmogorov theorem above. Therefore, there exists a probability space, (Ω,P,F )and measurable functions, ξ k : Ω→ R such that

P([

ξ i1 ∈ Fi1

]∩[ξ i2 ∈ Fi2

]· · ·∩

[ξ in ∈ Fin

])= µ i1···in (F1×·· ·×Fn)

= P([

ξ i1 ∈ Fi1

])· · ·P

([ξ in ∈ Fin

])which shows the random variables are independent as well as normal with mean 0 andvariance 1. This proves the Lemma.

A random variable X defined on a probability space (Ω,F ,P) is called Gaussian if

P([X ∈ A]) =1√

2πσ (v)2

∫A

e− 1

2σ(v)2(x−m(v))2

dx

for all A a Borel set in R. Therefore, for the probability space (X ,B (X) ,µ) it is natural tosay µ is a Gaussian measure if every x∗ in the dual space X ′ is a Gaussian random variable.That is, normally distributed.

Definition 61.8.3 Let µ be a measure defined on B (X) , the Borel sets of X , a separableBanach space. It is called a Gaussian measure if each of the functions in the dual spaceX ′ is normally distributed. As a special case, when X =U a separable real Hilberts space,µ is called a Gaussian measure if for each v ∈ U, the function u→ (u,v)U is normallydistributed. That is, denoting this random variable as v′, it follows for A a Borel set in R

λ v′ (A)≡ µ([

u : v′ (u) ∈ A])

=1√

2πσ (v)2

∫A

e− 1

2σ(v)2(x−m(v))2

dx

in case σ (v)> 0. In case σ (v) = 0

λ v′ ≡ δ m(v)

In other words, the random variables v′ for v ∈ U are all normally distributed on theprobability space (U,B (U) ,µ) .

Also recall the definition of the characteristic function of a measure.

Definition 61.8.4 The Borel sets in a topological space X will be denoted by B (X) . For aBorel probability measure µ defined on B (U) for U a real separable Hilbert space, defineits characteristic function as follows.

φ µ (u)≡ µ̂ (u)≡∫

Uei(u,v)dµ (v) (61.8.31)

More generally, if µ is a probability measure defined on B (X) where X is a separableBanach space, then the characteristic function is defined as

φ µ (x∗) = µ̂ (x∗)≡

∫U

eix∗(x)dµ (x)