79.4. STOCHASTIC INCLUSIONS WITHOUT UNIQUENESS ?? 2721

which is P measurable. Thus G is a measurable multifunction.Since (t,ω)→ G(t,ω) is measurable, there is a sequence {wn (t,ω)} of measurable

functions such that ∪∞n=1wn (t,ω) equals

G(t,ω) =

{w : ⟨w,u(t,ω)− y(t,ω)⟩ ≤ α (t,ω)+

1n

, (t,ω) /∈ Σ

}∩B(0,γ)

As shown above, there exists wt,ω in A(u(t,ω) , t,ω) as well as G(t,ω) . Thus there isa sequence of wr (t,ω) converging to wt,ω . Of course r will need to depend on t,ω . Since(t,ω)→ A(u(t,ω) , t,ω) is a P measurable multifunction, it has a countable subset of Pmeasurable functions {zk (t,ω)} which is dense in A(u(t,ω) , t,ω). Let Uk be defined as

Uk (t,ω)≡ ∪mB(

zm (t,ω) ,1k

)⊆ A(u(t,ω) , t,ω)+B

(0,

2k

)Now define A1k = {(t,ω) : w1 (t,ω) ∈Uk (t,ω)} . Then let

A2k = {(t,ω) /∈ A1k : w2 (t,ω) ∈Uk (t,ω)}

andA3k =

{(t,ω) /∈ ∪2

i=1Aik : w3 (t,ω) ∈Uk (t,ω)}

and so forth. Any (t,ω) ∈ Sγ must be contained in one of these Ark for some r since if notso, there would not be a sequence wr (t,ω) converging to wt,ω ∈ A(u(t,ω) , t,ω). TheseArk partition Sγ \Σ and each is measurable since the {zk (t,ω)} are measurable. Let

ŵk (t,ω)≡∞

∑r=1

XArk (t,ω)wr (t,ω)

Thus ŵk (t,ω) is in Uk (t,ω) for all (t,ω) ∈ Sγ and equals exactly one of the wm (t,ω) ∈G(t,ω).

Also, by construction, the ŵk (·, ·) are bounded in L∞(Sγ ;V ′

). Therefore, there is a sub-

sequence of these, still called ŵk which converges weakly to a function w in L2(Sγ ;V ′

).

Thus w is a weak limit point of co(∪∞

j=kŵ j

)for each k. Therefore, in the open ball

B(w, 1

k

)⊆ L2

(Sγ ;V ′

)with respect to the strong topology, there is a convex combination

∑∞j=k c jkŵ j (the c jk add to 1 and only finitely many are nonzero) which converges strongly

in L2(Sγ ;V ′

). Since G(t,ω) is convex and closed, this convex combination is in G(t,ω).

Off a set of P measure zero, we can assume this convergence of ∑∞j=k c jkŵ j as k→ ∞

happens pointwise a.e. for a suitable subsequence. However,∞

∑j=k

c jkŵ j (t,ω) ∈Uk (t,ω)⊆ A(u(t,ω) , t,ω)+B(

0,2k

).

Thus w(t,ω) ∈ A(u(t,ω) , t,ω) a.e. (t,ω) because A(u(t,ω) , t,ω) is a closed set. Sincew is the pointwise limit of measurable functions off a set of measure zero, it can be as-sumed measurable and for a.e. (t,ω), w(t,ω) ∈ A(u(t,ω) , t,ω)∩G(t,ω). Now denotethis measurable function wn. Then

wn (t,ω) ∈ A(u(t,ω) , t,ω) ,⟨wn (t,ω) ,u(t,ω)− y(t,ω)⟩ ≤ α (t,ω)+1n

a.e. (t,ω)