2250 CHAPTER 65. STOCHASTIC INTEGRATION

Then J∗ = ∑Li=1√

λ i(√

λ iei⊗ ei)

and JJ∗ = ∑Li=1 λ iei ⊗ ei = Q. Also, J is a Hilbert

Schmidt map into U from U0.

L

∑i=1

∥∥∥J(√

λ iei

)∥∥∥2

U=

L

∑i=1

∥∥∥√λ iei

∥∥∥2

U=

L

∑i=1

λ i < ∞

and so J is a Hilbert Schmidt mapping. In addition to this, from the construction, the spanof {ei}L

i=1 is dense in U0 and Jei = ei because

Jek =L

∑i=1

√λ i

(ei⊗

√λ iei

)(ek) = ek

√λ k

(ek,√

λ kek

)U0

= ek

so in fact J is just the injection map of U0 into U . Hence J−1 must also be the identitymap. Now we can let U1 =U with J the injection map. Thus, in this case, the elementaryfunctions Φn simply converge to Φ in

L2 ([a,T ]×Ω;L2 (JU0,H))

Note that∣∣∣∣J√λ iei

∣∣∣∣U =

√λ i whereas

∣∣∣∣√λ iei∣∣∣∣

Q1/2U = 1, and so J definitely does notpreserve norms. That is, the norm in U0 is not the same as the norm in U . Then everythingelse is the same. In particular

E

(∣∣∣∣∫ t

aΦdW

∣∣∣∣2H

)=∫ t

aE(||Φ||2L2(U0,H)

)ds.

65.7 A Short Comment On MeasurabilityIt will also be important to consider the composition of functions. The following is the mainresult. With the explanation of progressively measurable given, it says the composition ofprogressively measurable functions is progressively measurable.

Proposition 65.7.1 Let A : [a,T ]×V ×Ω → U where V,U are topological spaces andsuppose A satisfies its restriction to [a, t]×V ×Ω is B ([a, t])×B (V )×Ft measurable.This will be referred to as A is progressively measurable. Then if X : [a,T ]×Ω→ V isprogressively measurable, then so is the map

(t,ω)→ A(t,X (t,ω) ,ω)

Proof: Consider the restriction of this map to [a, t0]×Ω. For such (t,ω) , to say

A(t,X (t,ω) ,ω) ∈ O

for O a Borel set in U is to say that

X (t,ω) ∈{

v : (t,v,ω) ∈ A−1 (O) , t ≤ t0}≡ A−1 (O)tω

Consider the set {(t,ω) ∈ [a, t0]×Ω : X (t,ω) ∈ A−1 (O)tω

}