2628 CHAPTER 77. STOCHASTIC INCLUSIONS

so for v ∈U arbitrary,

⟨εξ ,u− v⟩= lim infh→0

(⟨εFuh (·,ω) ,u−uh⟩U ′,U + ⟨εFuh (·,ω) ,uh− v⟩U ′,U

)= lim inf

h→0⟨εFuh (·,ω) ,uh− v⟩U ′,U ≥ ⟨εFu,u− v⟩

and so, since v is an arbitrary element of U , it follows that ξ = F (u).Now consider the other term involving u∗h. Recall that u∗h ∈ A(τhuh,ω).

∥τhuh−uh∥V ≤ ∥τhuh− τhu∥V +∥τhu−u∥V≤ ∥uh−u∥V +∥τhu−u∥V

and both of these on the right converge to 0 thanks to continuity of translation and 77.5.43.Therefore,

limh→0⟨u∗h (·,ω) ,τhuh−u⟩V ′,V = 0.

It follows that

⟨u∗,u− v⟩V ′,V = lim infh→0

(⟨u∗h (·,ω) ,u− τhuh⟩V ′,V + ⟨u∗h (·,ω) ,τhuh− v⟩

)≥ lim inf

h→0⟨u∗h (·,ω) ,τhuh− v⟩ ≥ ⟨u∗ (v) ,u− v⟩

where u∗ (v)∈ A(u,ω). Then it follows that u∗ ∈ A(u,ω) because if not, then by separationtheorems, there would exist v such that

⟨u∗,u− v⟩V ′,V < ⟨w∗,u− v⟩V ′,Vfor all w∗ ∈ A(u,ω) which contradicts the above inequality. Thus, passing to the limit in77.5.37,

(B(ω)u(·,ω))′+ εFu(·,ω)+u∗ = f (·,ω) in U ′, (77.5.47)Bu(0,ω) = Bu0 (ω)

Here u∗ ∈A(u,ω) . Of course nothing is known about the measurability of u∗,u. All that hasbeen obtained in the above is a solution for each fixed ω . However, each of the functionsuh,u∗h is measurable. Also we have the estimate 77.5.39. By Theorem 77.2.10, there arefunctions û(·,ω) , û∗ (·,ω) and a subsequence with subscript h(ω) such that the followingweak convergences in V and V ′ take place

uh(ω) (·,ω)⇀ û(·,ω) , u∗h(ω) (·,ω)⇀ û∗ (·,ω)

such that the functions (t,ω)→ û(t,ω) ,(t,ω)→ û∗ (t,ω) are product measurable into Vand V ′ respectively. The above argument shows that for each ω , there is a further sub-sequence, still denoted with subscript h(ω) such that uh(ω) (·,ω)→ u(·,ω) weakly in Vand u∗h(ω) (·,ω)→ u∗ (·,ω) weakly in V ′ such that (u(·,ω) ,u∗ (·,ω)) is a solution to theevolution equation for each ω . By uniqueness of limits, u(·,ω) = û(·,ω) , similar for û∗.Thus this solution which is defined for each ω has a representative for each ω such that theresulting functions of t,ω are product measurable into V,V ′ respectively. This proves thefollowing lemma.