2652 CHAPTER 77. STOCHASTIC INCLUSIONS

where limµ,ν→0 ε (µ,ν) = 0. It follows from 77.8.97

lim supµ,ν→0

(∫ T

0

⟨zµ − zν ,uµ −uν

⟩ds+ ε (µ,ν)

)= lim sup

µ,ν→0

(∫ T

0

⟨zµ − zν ,uµ −uν

⟩ds)≤ 0

From Lemma 77.8.4,lim sup

µ→0

⟨zµ ,uµ −u

⟩V ′,V ≤ 0

By the limit condition for A(·,ω) , for each v ∈ V , there exists z(v) ∈ Au such that

lim infµ→0

⟨zµ ,uµ − v

⟩= lim inf

µ→0

(⟨zµ ,uµ −u

⟩+⟨zµ ,u− v

⟩)= ⟨z,u− v⟩ ≥ ⟨z(v) ,u− v⟩

Since A(u,ω) is convex and closed, separation theorems imply that z ∈ Au. Return to theequation solved. (

Buµ

)′+ zµ +gµ = f

Then act on uµ −u and use monotonicity arguments to write⟨(Bu)′ ,uµ −u

⟩V ′,V +

⟨zµ ,uµ −u

⟩V ′,V +

⟨gµ ,uµ −u

⟩V ′,V ≤

⟨f ,uµ −u

⟩V ′,V (77.8.98)

Then it was shown above that

0≥ lim supµ→0

⟨zµ ,uµ −u

⟩V ′,V ≥ lim inf

µ→0

⟨zµ ,uµ −u

⟩V ′,V ≥ ⟨z(u) ,u−u⟩V ′,V = 0

and so, from 77.8.98,

limµ→0

⟨gµ ,uµ −u

⟩V ′,V = lim

µ→0

⟨gµ ,Jµ uµ −u

⟩V ′,V = 0

and solimµ→0

⟨gµ ,Jµ uµ

⟩V ′,V = ⟨g,u⟩V ′,V

Now let [a,b] ∈ G (G) . Then

⟨b−g,a−u⟩= limµ→0

⟨b−gµ ,a− Jµ uµ

⟩≥ 0

because gµ ∈ G(Jµ uµ

). Since G is maximal monotone, it follows that [u,g] ∈ G (G).

This has shown that for each ω fixed, and every sequence of solutions to the integralequation

{uµ

}, each function

{Buµ

}being product measurable by Theorem 77.5.7, there

exists a subsequence which converges to a solution u to the integral equation. In particular,t → Bu(t) is weakly continuous into V ′. Then by the fundamental measurable selectiontheorem, Theorem 77.2.10, there exists a product measurable function ū(t,ω) with valuesin V weakly continuous in t and a sequence depending on ω,

{uµ(ω)

}such that for each