61.9. ABSTRACT WIENER SPACES 2035

However, H is also equal to the countable union of the sets,

An ≡{

x ∈ H : ((x,e1)H , · · · ,(x,ean)H) ∈ B(0,n)}

where an→ ∞.

ν (An) ≡ 1(√2π)an

∫B(0,n)

e−12 |t|

2dt

≤ 1(√2π)an

∫ n

−n· · ·∫ n

−ne−|t|

2/2dt1 · · ·dtan

=

(∫ n−n e−x2/2dx√

2π

)an

Now pick an so large that the above is smaller than 1/2n+1. This can be done because forno matter what choice of n, ∫ n

−n e−x2/2dx√

2π< 1.

Then∞

∑n=1

ν (An)≤∞

∑n=1

12n+1 =

12.

This proves the proposition and shows something else must be done to get a countablyadditive measure from ν .

However, let µ (C) ≡ νM (C) where C is a cylinder set of the form C = B+M⊥ for Ma finite dimensional subspace.

Proposition 61.9.8 µ is finitely additive on C the algebra of cylinder sets.

Proof: LetA≡ {x ∈ H : ((x,e1) , · · · ,(x,en)) ∈ E} ,

B≡ {x ∈ H : ((x, f1) , · · · ,(x, fm)) ∈ F}

be two disjoint cylinder sets. Then writing them differently as was done earlier they are

{x ∈ H : ((x,e1) , · · · ,(x,en) ,(x, f1) , · · · ,(x, fm)) ∈ E×Rm}

and{x ∈ H : ((x,e1) , · · · ,(x,en) ,(x, f1) , · · · ,(x, fm)) ∈ Rn×F}

respectively. Hence the two sets E ×Rm,Rn×F must be disjoint. Then the definitionyields µ (A∪B) = µ (A)+µ (B). This proves the proposition.

Definition 61.9.9 Let H be a separable Hilbert space and let ||·|| be a norm defined on Hwhich has the following property. Whenever {en} is an orthonormal sequence of vectorsin H and F ({en}) consists of the set of all orthogonal projections onto the span of finitely