77.4. MEASURABLE APPROXIMATE SOLUTIONS 2619
Proof: Let Bn (ω)≡ kn (ω)B where {kn (ω)} is an increasing sequence of simple func-tions converging pointwise to k (ω). Replace B(ω) with Bn (ω) . Then define
⟨Knu,v⟩ ≡∫ T
0⟨Lnu,v⟩ds+ ⟨Bu,v⟩(0)
where Ln is defined asLnu = (Bn (ω)u)′
for Bn having values in L (W,W ′) such that Bn (ω)→B(ω) and each of these is self adjointand nonnegative. Let un be the solution to the above initial value problem
⟨Knun,v⟩+Fun = ⟨ fn,v⟩+ ⟨Bv(0) ,u0n⟩
in which u0n and fn are simple functions converging to u0 and f in W and V ′ respectivelyfor each ω . Thus these have constant values in V ′ or W on finitely many measurable subsetsof Ω. Since Bn is constant on measurable sets, it follows that un (·,ω) is also a constantelement of V on each of finitely many measurable sets. Hence un (·,ω) is measurable intoV . Then fixing ω, and letting v = un,
12[⟨Bun,un⟩(T )+ ⟨Bun,un⟩(0)]+
∫ T
0∥un∥p
V ds =∫ T
0⟨ fn,un⟩ds+ ⟨Bv(0) ,u0n⟩
Thus, since F is bounded, one obtains an inequality of the form
∥un∥V +∥∥(Bn (ω)un)
′∥∥V ′ ≤C
Then there is a subsequence such that
un→ u weakly in V
Bn (ω)un→ B(ω)u weak ∗ in L∞([0,T ] ;V ′
)Bn (ω)un→ B(ω)u weakly in V ′
Fun→ ξ weakly in V ′
(Bn (ω)un)′→ (B(ω)u)′ weakly in V ′
Also, suppressing the dependence on ω,
(Bnun)(t) = Bnu0n +∫ t
0(Bnu)′ (s)ds
and so in fact,(Bnun)(t)→ (Bu)(t) in V ′ for each t
Also, ⟨(Bnun)
′ ,un−u⟩V ′,V + ⟨Fun,un−u⟩V ′,V = ⟨ fn,un−u⟩V ′,V⟨
kn (ω)(Bun)′ ,un−u
⟩V ′,V + ⟨Fun,un−u⟩V ′,V = ⟨ fn,un−u⟩V ′,V