2602 CHAPTER 77. STOCHASTIC INCLUSIONS

Now here is the proof of the theorem.Proof of Theorem 77.2.4 By assumption, there exists a further subsequence, still denotedby n(ω), such that, in addition to 77.2.3, the weak limit

limn(ω)→∞

XNC (ω)un(ω) (·,ω) = u(·,ω)

exists in Lp′ ([0,T ] ;V ′) such that t → u(t,ω) is weakly continuous into V ′. Then, 77.2.3also holds for this further subsequence and in addition,

mk

∫ t

lmk (t)

⟨φ rk

,u(s,ω)⟩

Vds

= limn(ω)→∞

mk

∫ t

lmk (t)

⟨φ rk

,XNC (ω)un(ω) (s,ω)⟩

Vds

= γk (t,ω) .

Letting φ ∈ D be given, there exists a subsequence, denoted by k, such that mk → ∞ andφ rk

= φ . Recall(

mk,φ rk

)denoted an enumeration of the pairs (m,φ) ∈ N×D . Then,

passing to the limit and using the assumed continuity of s→ u(s,ω) , the left-hand sideof this equality converges to ⟨φ ,u(s,ω)⟩V and so the right-hand side, γk (t,ω), must alsoconverge and for each ω . Since the right-hand side is a product measurable function of(t,ω) , it follows that the pointwise limit is also product measurable. Hence, (t,ω)→⟨φ ,u(t,ω)⟩V is product measurable for each φ ∈D . Since D is a dense set, it follows that(t,ω)→ ⟨φ ,u(t,ω)⟩V is P measurable for all φ ∈ V and so by the Pettis theorem, [127],(t,ω)→ u(t,ω) is P measurable into V ′.

Actually, one can say more about the measurability of the approximating sequence andin fact, we can obtain one for which ω → un(ω) (t,ω) is also F measurable.

Lemma 77.2.9 Suppose that un(ω)→ u weakly in Lp′ ([0,T ] ;V ′), where u is product mea-surable, and

{un(ω)

}is a subsequence of {un}, such that there exists a set of measure zero

N ⊆Ω andsup

t∈[0,T ]∥un (t,ω)∥V ′ <C (ω) , for ω /∈ N.

Then, there exists a subsequence of {un}, denoted as{

uk(ω)

}, such that uk(ω)→ u weakly

in Lp′ ([0,T ] ;V ′), ω→ k (ω) is F measurable, and ω→ uk(ω) (t,ω) is also F measurable,for each ω /∈ N.

Proof: Assume that f ,g ∈ Lp′ ([0,T ] ;V ′) and let {φ k} be a countable dense subsetof Lp ([0,T ] ;V ). Then, a bounded set in Lp′ ([0,T ] ;V ′) with the weak topology can beconsidered a complete metric space using the metric

d ( f ,g)≡∞

∑j=1

2− j |⟨φ k, f −g⟩|1+ |⟨φ k, f −g⟩|

.

Now, let k (ω) be the first index of {un} that is at least as large as k and such that

d(XNC (ω)uk(ω),u

)≤ 2−k.