76.3. THE EXISTENCE OF APPROXIMATE SOLUTIONS 2547

It follows that ∫Ω

⟨g(ω) ,

∫ T

0

(f − f̄

)ψ (t)dt

⟩dP = 0

This is true for every such g ∈ L∞ (Ω,X ′) , and so for a fixed ψ ∈C ([0,T ]) and the Rieszrepresentation theorem, ∫

Ω

∥∥∥∥∫ T

0

(f − f̄

)ψ (t)dt

∥∥∥∥X

dP = 0

Therefore, there exists Nψ such that if ω /∈ Nψ , then

∫ T

0

(f − f̄

)ψ (t)dt = 0

Enlarge N, the exceptional set to also include ∪ψ∈DNψ where D is a countable dense subsetof C ([0,T ]). Therefore, if ω /∈N, then the above holds for all ψ ∈C ([0,T ]). It follows thatfor such ω, f (t,ω) = f̄ (t,ω) for a.e. t. Therefore, f (·,ω) = f̄ (·,ω) in Lα (0,T,X) for allω /∈ N.

Then one can obtain the following existence theorem using a technique of Bardos andBrezis [14].

Lemma 76.3.5 Let q ∈ V and let the conditions 76.3.14 - 76.3.17 be valid. Let f ∈ V ′

be given. Then for each ω off a set of measure zero, there exists u(·,ω) ∈ Vω such that(Bu)′ (·,ω) ∈ V ′ω and

Bu(0,ω) = 0

and also the following equation holds in V ′ω for a.e. ω

(Bu)′ (·,ω)+ Ā(ω)(·,u(·,ω)) = f (·,ω)

In addition to this, it can be assumed that (t,ω)→ u(t,ω) is progressively measurable intoV . That is, for each ω off a set of measure zero, t → u(t,ω) can be modified on a set ofmeasure zero in [0,T ] such that the resulting u is progressively measurable.

Proof: Consider the equation

LhBu+ Āu =1h(I− τh)(Bu)+ Āu = f in V ′ (76.3.18)

By Proposition 76.2.4 and Theorem 76.2.6, there exists a solution to the above equation ifthe left side is coercive. However, it was shown above in the computations leading to 76.2.5that Lh ◦B is monotone. Hence the coercivity follows right away from Lemma 76.3.2.

Thus 76.3.18 holds in V ′. It follows that, indexing the solution by h,

∫Ω

∫ T

0

∥∥∥∥1h(I− τh)(Buh)+ Āuh− f

∥∥∥∥p′

V ′dtdP = 0