2546 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

Lemma 76.3.4 Suppose fn is progressively measurable and converges weakly to f̄ in

Lα ([0,T ]×Ω,X ,B ([0,T ])×FT ) , α > 1

where X is a reflexive separable Banach space. Also suppose that for each ω /∈ N a set ofmeasure zero,

fn (·,ω)→ f (·,ω) weakly in Lα (0,T,X)

Then there is an enlarged set of measure zero, still denoted as N such that for ω /∈ N,

f̄ (·,ω) = f (·,ω) in Lα (0,T,X) .

Also f̄ is progressively measurable.

Proof: By the Pettis theorem, f̄ is progressively measurable. Letting

φ ∈ Lα ′ ([0,T ]×Ω,X ′,B ([0,T ])×FT),

it is known that for a.e. ω,∫ T

0⟨φ (t,ω) , fn (t,ω)⟩dt→

∫ T

0⟨φ (t,ω) , f (t,ω)⟩dt

Therefore, the function of ω on the right is at least FT measurable. Now let

g ∈ L∞(Ω,X ′,FT

)and let ψ ∈C ([0,T ]). Then for 1 < p≤ α,∫

Ω

∣∣∣∣∫ T

0⟨g(ω)ψ (t) , fn (t,ω)⟩dt

∣∣∣∣p dP

≤ C (T )∫

Ω

∥g∥pL∞(Ω,X ′)

∫ T

0|ψ (t)|p ∥ fn (t,ω)∥p

X dtdP

≤ C (T,g,ψ)∫

Ω

∫ T

0∥ fn (t,ω)∥p

X dtdP≤C < ∞

for some C. Since∫ T

0 ⟨g(ω)ψ (t) , fn (t,ω)⟩dt is bounded in Lp (Ω) independent of nbecause

∫Ω

∫ T0 ∥ fn (t,ω)∥p

X dtdP is given to be bounded, it follows that the functions

ω →∫ T

0⟨g(ω)ψ (t) , fn (t,ω)⟩dt

are uniformly integrable and so it follows from the Vitali convergence theorem that∫Ω

∫ T

0⟨g(ω)ψ (t) , fn (t,ω)⟩dtdP→

∫Ω

∫ T

0⟨g(ω)ψ (t) , f (t,ω)⟩dtdP

But also from the assumed weak convergence to f̄∫Ω

∫ T

0⟨g(ω)ψ (t) , fn (t,ω)⟩dtdP→

∫Ω

∫ T

0

⟨g(ω)ψ (t) , f̄ (t,ω)

⟩dtdP