77.5. THE MAIN RESULT 2623

Proof: Let ω→ u(ω) be measurable into V and let un (ω)→ u(ω) in V where un is asimple function

un (ω) =mn

∑k=1

cnkXEn

k(ω) , the En

k disjoint, Ω = ∪kEnk ,

each cnk being in V . Then by assumption, there is a measurable selection for ω→ A

(cn

k ,ω)

denoted as ω→ ynk (ω) . Thus ω→ yn

k (ω) is measurable into V ′ and ynk (ω) ∈ A

(cn

k ,ω)

forall ω ∈Ω. Then consider

yn (ω) =mn

∑k=1

ynk (ω)XEn

k(ω)

It is measurable and for ω ∈ Enk it equals yn

k (ω) ∈ A(cn

k ,ω)= A(un (ω) ,ω) . Thus yn is

a measurable selection of ω → A(un (ω) ,ω) . By the estimates, for each ω these yn (ω)lie in a bounded subset of V ′. The bound might depend on ω of course. By Theorem77.2.10 and Lemma 77.4.2 there is a measurable into V ′ function ω → y(ω) and a sub-sequence for each ω,yn(ω) (ω) such that yn(ω) (ω)→ y(ω) weakly in V ′. By the Pettistheorem, y is measurable into V ′. Where is y(ω)? If y(ω) /∈ A(u(ω) ,ω) , then therewould exist z(ω) ∈ V such that ⟨y(ω) ,z⟩ > r > ⟨w,z⟩ for all w ∈ A(u(ω) ,ω). Let O ={w ∈ V ′ such that r > ⟨w,z⟩} . Then O contains A(u(ω) ,ω) and is a weakly open set. Itfollows from the upper semicontinuity assumption that yn(ω) (ω) ∈ O for all n(ω) largeenough. Thus r >

⟨yn(ω) (ω) ,z

⟩. But by weak convergence,

⟨y(ω) ,z⟩> r ≥ limn(ω)→∞

⟨yn(ω) (ω) ,z

⟩= ⟨y(ω) ,z⟩

contradicting y(ω) /∈ A(u(ω) ,ω). Hence y(ω)∈ A(u(ω) ,ω) and ω→ y(ω) is a measur-able selection.

In fact, this is just a special case of a general result in the next theorem. It says essen-tially that having a measurable selection is preserved when going from constant to measur-able functions. In this theorem, V is a reflexive separable Banach space. This is difficult toshow for measurable multifunctions. It is Theorem 48.3.1 and was proved earlier. A proofis given here also.

Theorem 77.5.3 Suppose ω → A(u,ω) has a measurable selection in V ′ for u a given el-ement of V not dependent on ω and for each ω,A(u,ω) is a closed, convex set in V ′ andA(·,ω) is bounded. Also suppose u→ A(u,ω) is upper semicontinuous from the strongtopology of V to the weak topology of V ′. That is, if un → u in V strongly, then if O is aweakly open set containing A(u,ω) , it follows that A(un,ω) ∈ O for all n large enough.Conclusion: whenever ω → u(ω) is measurable into V , it follows that there is a measur-able selection for ω → A(u(ω) ,ω) into V ′.

Proof: Let ω → u(ω) be measurable into V and let un (ω)→ u(ω) in V where un is asimple function

un (ω) =mn

∑k=1

cnkXEn

k(ω) , the En

k disjoint, Ω = ∪kEnk ,