76.4. THE GENERAL CASE 2563

ξ in Vω . Next note that the set where ξ n is not a Cauchy sequence is a progressivelymeasurable set. It equals

∪n∩m∪k,l≥m

[(t,ω) : ∥ξ l (t,ω)−ξ k (t,ω)∥> 1

n

]≡ S

Now for p > 0

limm→∞

P

(supp>0

∥∥ξ m+p−ξ m

∥∥Vω

> ε

)= 0

This is because of the convergence of ξ n to ξ in Vω . Therefore, there is a subsequence stillcalled ξ n such that

P

(supp>0

∥∥ξ n+p−ξ n

∥∥Vω

> 2−n

)< 2−n

and so there is an enlarged set of measure zero, still denoted as N such that all of the aboveconsiderations hold for ω /∈ N and also for ω /∈ N,

supp>0

∥∥ξ n+p−ξ n

∥∥Vω

≤ 2−n

for all n large enough. Now let S defined above, correspond to this particular subsequence.Let S (ω) be those t such that (t,ω)∈ S. Then S (ω) is a set of measure zero for each ω /∈Nbecause the above inequality implies that t → ξ n (t,ω) is a Cauchy sequence off a set ofmeasure zero which by definition is S (ω). Then consider {ξ n (t,ω)XSC (t,ω)}. For eachω off N, this converges for all t. Thus it converges pointwise to a function ξ̄ which mustbe progressively measurable. However, t→ ξ̄ (t,ω) must also equal t→ ξ (t,ω) in Vω bythe above construction. Therefore, we can assume without loss of generality that ξ is itselfprogressively measurable.

From the weak convergence of un to u in Vω ,

Bun→ Bu weakly in V ′ω

and so(λB+A(ω))un→ λBu+ξ weakly in V ′ω

Now the above convergences and the integral equation imply that off the exceptionalset N, for each t

Bun (t)→ Bu(t) weakly in V ′

From a generalization of standard theorems in Hilbert space, stated in Lemma 76.2.1 thereexist vectors {ei} ⊆V such that

⟨Bun (t) ,un (t)⟩=∞

∑i=1|⟨Bun (t) ,ei⟩|2

Hence

lim infn→∞⟨Bun (t) ,un (t)⟩ ≥

∞

∑i=1

lim infn→∞|⟨Bun (t) ,ei⟩|2