2608 CHAPTER 77. STOCHASTIC INCLUSIONS

ψ to the closed subspace W is at least 1/5. Now C ,∪{C ∪{ψ}} would violate maximalityof C . Hence W =V ′. Now it follows that C must be countable since otherwise, V ′ wouldfail to be separable. Let M be the rational linear combinations of D . It must be dense in V ′.Note that linear combinations of the φ i are uniquely determined because none is a linearcombination of the others. Now define a linear mapping on M which makes sense for (t,ω)on a certain set.

Definition 77.2.15 Let E be those points (t,ω) such that the following limit exists for eachφ ∈D

Λ(t,ω)φ ≡ limmφ→∞

mφ

∫ t

lmφ(t)⟨φ ,v(s,ω)⟩ds

The set of points where the limit of measurable functions exists is always measurable so Eis a measurable set. Extend this mapping linearly. That is, for ψ ∈M,ψ ≡ ∑i aiφ i,

Λ(t,ω)ψ ≡∑i

aiΛ(t,ω)φ i = ∑i

ai

(lim

mφ i→∞mφ i

∫ t

lmφ i(t)⟨φ i,v(s,ω)⟩ds

)Thus (t,ω)→ Λ(t,ω)ψ is product measurable, being the sum of limits of product measur-able functions. Let G denote those (t,ω) in E such that there exists a constant C (t,ω) suchthat for all ψ ∈M,

|Λ(t,ω)ψ| ≤C (t,ω)∥ψ∥

Lemma 77.2.16 G is product measurable.

Proof: This follows from the formula

E ∩GC = ∩n∪ψ∈M {(t,ω) ∈ E : |Λ(t,ω)ψ|> n∥ψ∥}

which is clearly product measurable because (t,ω)→ Λ(t,ω)ψ is. Thus, since E is mea-surable, it follows that E ∩G = G is also.

For (t,ω) ∈ G,Λ(t,ω) has a unique extension to all of V , the dual space of V ′, stilldenoted as Λ(t,ω). By the Riesz representation theorem, for (t,ω) ∈ G, there existsu(t,ω) ∈V,

Λ(t,ω)ψ = ⟨ψ,u(t,ω)⟩V ′,VThus (t,ω)→XG (t,ω)u(t,ω) is product measurable by the Pettis theorem. Let u = 0 offG. We know G is product measurable. For each ω,{t : (t,ω) ∈ G} has full measure. Thisinvolves the fundamental theorem of calclulus.

Fix ω . By the fundamental theorem of calculus,

limm→∞

m∫ t

lm(t)v(s,ω)ds = v(t,ω) in V

for a.e. t say for all t /∈ N (ω) ⊆ [0,T ]. Of course we do not know that ω → v(t,ω)is measurable. However, the existence of this limit for t /∈ N (ω) implies that for everyφ ∈V ′,

limm→∞

∣∣∣∣m∫ t

lm(t)⟨φ ,v(s,ω)⟩ds

∣∣∣∣≤C (t,ω)∥φ∥