2622 CHAPTER 77. STOCHASTIC INCLUSIONS

Here is an alternate limit condition. Let U be a Banach space dense in V and that ifui ⇀ u in VI and u∗i ∈ A(ui) with u∗i ⇀ u∗ in V ′I and t→ Bui (t) is continuous and

supi

supt ̸=s

∥Bui (t)−Bui (s)∥U ′|t− s|α

≤C (77.5.31)

then iflim sup

i→∞

⟨u∗i ,ui−u⟩V ′I ,VI≤ 0 (77.5.32)

it follows that for all v ∈ VI , there exists u∗(v) ∈ Au such that

lim infi→∞⟨u∗i ,ui− v⟩V ′I ,VI

≥ ⟨u∗ (v) ,u− v⟩V ′I ,VI(77.5.33)

This alternate condition is implied by 77.5.30 but the conditions under which either con-dition holds are likely to depend on some sort of compactness which will be useable foreither limit condition. Technically if you assume this alternate condition, you are assumingmore, but I don’t have any examples to show that it would be actually assuming more.

For ω → u(·,ω) measurable into V ,

ω → A(u(·,ω) ,ω) has a measurable selection into V ′. (77.5.34)

This last condition means there is a function ω → u∗ (ω) which is measurable into V ′

such that u∗ (ω) ∈ A(u(·,ω) ,ω) . This is assured to take place if the following standardmeasurability condition is satisfied for all O open in V ′:

{ω : A(u(·,ω) ,ω)∩O ̸= /0} ∈F (77.5.35)

See for example [70], [10]. Our assumption is implied by this one but they are not equiv-alent. Thus what is considered here generalizes an assumption that ω → A(u(·,ω) ,ω) isset valued measurable.

Note that this condition would hold if u→ A(t,u,ω) is bounded and pseudomono-tone as a single valued map from V to V ′ and (t,ω)→ A(t,u,ω) is product measurableinto V ′. One would use the demicontinuity of u → A(·,u,ω) which comes from thepseudo monotone and bounded assumption and consider a sequence of simple functionsun (t,ω)→ u(t,ω) in V for u measurable, each u(·,ω) being in V , Then the measurabil-ity of A(t,un,ω) would attach to A(t,u,ω) in the limit. More generally, here is a usefullemma. It is about preserving the existence of a measurable representative under the as-sumption that the values are closed and convex.

Lemma 77.5.2 Suppose ω → A(u,ω) has a measurable selection in V ′ for u a givenelement of V not dependent on ω and for each ω,A(u,ω) is a closed bounded, convex setin V ′. Also suppose u→ A(u,ω) is upper semicontinuous from the strong topology of Vto the weak topology of V ′. That is, if un→ u in V strongly, then if O is a weakly open setcontaining A(u,ω) , it follows that A(un,ω) ∈ O for all n large enough. Then whenever uis measurable into V , it follows that there is a measurable selection for ω → A(u(ω) ,ω)into V ′.