2034 CHAPTER 61. PROBABILITY IN INFINITE DIMENSIONS

where from the definition,

(θ ∗eθ es, t) = ∑i

ti

(∑

js je j,ei

)= ∑

itisi = (s, t)

and so θ∗eθ e is the identity on Rn and similar reasoning yields θ eθ

∗e is the identity on

θ e (Rn). Then using the change of variables formula and the fact |det(L)|= 1,

1

(2π)n/2 (det(θ ∗eQθ e))1/2

∫F

e−12 t∗θ∗eQ−1θ etdt

=1

(2π)n/2 (det(θ ∗eQθ e))1/2

∫G

e−12 s∗L∗θ∗eQ−1θ eLsds

=1

(2π)n/2 (det(θ∗f Qθ f

))1/2

∫G

e−12 s∗θ∗f θ eθ

∗eQ−1θ eθ

∗eθ f sds

=1

(2π)n/2 (det(θ∗f Qθ f

))1/2

∫G

e−12 s∗θ∗f Q−1θ f sds

where part of the justification is as follows.

det(θ∗f Qθ f

)= det

(θ∗f θ eθ

∗eQθ eθ

∗eθ f)

= det(θ∗f θ e)

det(θ ∗eQθ e)det(θ∗eθ f)

= det(θ ∗eQθ e)

becausedet(θ∗f θ e)

det(θ∗eθ f)= det

(θ∗f θ eθ

∗eθ f)= det

(θ∗f θ f)= 1.

This proves the lemma.It would be natural to try to extend ν to the σ algebra determined by C and obtain a

measure defined on this σ algebra. However, this is always impossible in the case whereQ = I.

Proposition 61.9.7 For Q = I, ν cannot be extended to a measure defined on σ (C ) when-ever H is infinite dimensional.

Proof: Let {en} be a complete orthonormal set of vectors in H. Then first note that His a cylinder set.

H = {x ∈ H : (x,e1) ∈ R}

and soν (H) =

1√2π

∫R

e−12 t2

dt = 1.