2032 CHAPTER 61. PROBABILITY IN INFINITE DIMENSIONS

combinations is dense in H ′ and so the conclusion of the lemma follows from Lemma59.4.2 on Page 1870. This proves the lemma.

Also note that the mapping

x→ ((x,e1) , · · · ,(x,en))

is a σ (C ) measurable map. Restricting it to span(e1, · · · ,en) , it is Borel measurable. Nextis a definition of a Gaussian measure defined on C . While this is what it is called, it is afake measure in general because it cannot be extended to a countably additive measure onσ (C ). This will be shown below.

Definition 61.9.5 Let Q ∈L (H,H) be self adjoint and satisfy

(Qx,x)> 0

for all x ∈ H,x ̸= 0. Define ν on the cylinder sets, C by the following rule. For {ek}nk=1 an

orthonormal set in H,

ν ({x ∈ H : ((x,e1) , · · · ,(x,en)) ∈ F})

≡ 1

(2π)n/2 (det(θ ∗Qθ))1/2

∫F

e−12 t∗θ∗Q−1θ tdt.

where here

θ t≡n

∑i=1

tiei.

Note that the cylinder set is of the form

θF + span(e1, · · · ,en)⊥ .

Thus if B+M⊥ is a typical cylinder set, choose an orthonormal basis for M,{ek}nk=1 and

do the above definition with F = θ−1B.

To see this last claim which is like what was done earlier, let

((x,e1) , · · · ,(x,en)) ∈ F.

Then θ ((x,e1) , · · · ,(x,en)) = ∑i (x,ei)ei = Px and so

x = x−Px+Px = x−Px+θ ((x,e1) , · · · ,(x,en))

∈ θF + span(e1, · · · ,en)⊥

Thus{x ∈ H : ((x,e1) , · · · ,(x,en)) ∈ F} ⊆ θF + span(e1, · · · ,en)

⊥

To see the other inclusion, if t ∈ F and y ∈ span(e1, · · · ,en)⊥ , then if x = θ t, it follows

ti = (x,ei)

and so ((x,e1) , · · · ,(x,en)) ∈ F. But (y,ek) = 0 for all k and so x+ y is in

{x ∈ H : ((x,e1) , · · · ,(x,en)) ∈ F} .