2638 CHAPTER 77. STOCHASTIC INCLUSIONS

there would be a subsequence, still denoted as un (ω) which converges to some q(ω) ̸=projK (ω) u(ω). Then

λ (ω) = limn→∞∥u(ω)−un (ω)∥ ≥ ∥u(ω)−q(ω)∥

because convex and lower semicontinuous is weakly lower semicontinuous. But this im-plies q(ω) = projK (ω) (u(ω)) because the projection map is well defined thanks to strictconvexity of the norm used. This is a contradiction. Hence projK (ω) u(ω) = limn→∞ un (ω)and so is a measurable function. It follows that ω→ P(u(ω) ,ω) is measurable into V .

The following corollary is now immediate.

Corollary 77.6.3 Suppose A(·,ω) is monotone hemicontinuous bounded, single valued,and coercive as a map from V to V ′. Suppose also that for ω → u(ω) measurable intoV , it follows that ω → A(u(ω) ,ω) is measurable into V ′. Let K (ω) be a closed convexsubset of V containing 0 and ω→K (ω) is a set valued measurable multifunction. Let B∈L (W,W ′) be self adjoint and nonnegative as above. Let there be a regularizing sequence{ui} for each u ∈K satisfying Bui (0) = 0,(Bui)

′ ∈ V ′,ui ∈K ,

lim supi→∞

⟨(Bui)

′ ,ui−u⟩≤ 0

Then for each ω, there exists a solution to⟨(Bv)′ ,u− v

⟩+ ⟨A(u(·,ω) ,ω) ,u(·,ω)− v⟩ ≤ ⟨ f (·,ω) ,u(·,ω)− v⟩

valid for all v∈K (ω) with (Bv)′ ∈V ′,Bv(0) = 0, and (t,ω)→ u(t,ω) , is B ([0,T ])×Fmeasurable.

Proof: The proof is identical to the above. One obtains a measurable solution to 77.6.64in which P is replaced with P(·,ω) . Then one proceeds in exactly the same steps as beforeand finally uses Theorem 77.2.10 to obtain the measurability of a solution to the variationalinequality.

What does it mean for u(ω) ∈K (ω) for each ω? It means that there is a sequence ofthe wn

{wn(ω)

}such that each wn is measurable into V implying that for each ω there is a

representative t→wn (t,ω) such that the resulting (t,ω)→wn (t,ω) is product measurableand

∥∥u(·,ω)−wn(ω) (·,ω)∥∥

V→ 0. Thus there is no reason to think that (t,ω)→ u(t,ω)

is product measurable. The message of the above corollary says that nevertheless, there isa measurable solution to the variational inequality.

77.7 Progressively Measurable SolutionsIn the context of uniqueness of the evolution initial value problem for fixed ω, one canprove theorems about progressively measurable solutions fairly easily. First is a definitionof the term progressively measurable.

Definition 77.7.1 Let Ft be an increasing in t set of σ algebras of sets of F . Thus eachFt is a σ algebra and if s≤ t, then Fs ≤Ft . This set of σ algebras is called a filtration.A set S⊆ [0,T ]×Ω is called progressively measurable if for every t ∈ [0,T ] ,

S∩ [0, t]×Ω ∈B ([0, t])×Ft