2610 CHAPTER 77. STOCHASTIC INCLUSIONS

let the functions t → yn (t,ω) be in V ′ and (t,ω)→ yn (t,ω) is P measurable into V ′.Suppose there is a set of measure zero N ⊆Ω such that if ω /∈ N, then

supt∈[0,T ]

∥yn (t,ω)∥V ′ +∥un (·,ω)∥V ≤C (ω) ,

for all n. (Thus, by weak compactness, for each ω, each subsequence of {un} has a furthersubsequence that converges weakly in V to v(·,ω) ∈ V . (v not known to be P measur-able)) Suppose that each subsequence of {yn (·,ω)} has a subsequence which convergesweakly in V ′ to z(·,ω) ∈ V ′ such that the function t → z(t,ω) is weakly continuous intoV ′.

Then, there exist product measurable functions u,y such that t → u(t,ω)is in V , t →y(t,ω) is weakly continuous into V ′ and for each ω /∈ N, a subsequence of N denoted by{n(ω)} such that un(ω) (·,ω)→ u(·,ω) weakly in V and yn(ω) (·,ω) converges weakly toy(·,ω) in V ′.

Note that the conclusion of the proposition holds if p = 1 and V = L1 ([0,T ] ,V ).Here is something else about being measurable into V or V ′. Such functions have

representatives which are product measurable.

Lemma 77.2.18 Let f (·,ω) ∈ V ′. Then if ω → f (·,ω) is measurable into V ′, it followsthat for each ω, there exists a representative f̂ (·,ω)∈V ′, f̂ (·,ω) = f (·,ω) in V ′ such that(t,ω)→ f̂ (t,ω) is product measurable. If f (·,ω) ∈ V ′ and (t,ω)→ f (t,ω) is productmeasurable, then ω → f (·,ω) is measurable into V ′. The same holds replacing V ′ withV .

Proof: If a function f is measurable into V ′, then there exist simple functions fn

limn→∞∥ fn (ω)− f (ω)∥V ′ = 0, ∥ fn (ω)∥ ≤ 2∥ f (ω)∥V ′ ≡C (ω)

Now one of these simple functions is of the form ∑Mi=1 ciXEi (ω) where ci ∈ V ′. There-

fore, there is no loss of generality in assuming that ci (t) = ∑Nj=1 di

jXFj (t) where dij ∈ V ′.

Hence we can assume each fn is product measurable into B (V ′)×F . Then by Theorem77.2.10, there exists f̂ (·,ω) ∈ V ′ such that f̂ is product measurable and a subsequencefn(ω) converging weakly in V ′ to f̂ (·,ω) for each ω . Thus fn(ω) (ω)→ f (ω) strongly inV ′ and fn(ω) (ω)→ f̂ (ω) weakly in V ′. Therefore, f̂ (ω) = f (ω) in V ′ and so it can beassumed that if f is measurable into V ′ then for each ω, it has a representative f̂ (ω) suchthat (t,ω)→ f̂ (t,ω) is product measurable.

If f is product measurable into V ′ and each f (·,ω) ∈ V ′, does it follow that f is mea-surable into V ′? By measurability, f (t,ω) = limn→∞ ∑

mni=1 cn

i XEni(t,ω) = limn→∞ fn (t,ω)

where Eni is product measurable and we can assume ∥ fn (t,ω)∥V ′ ≤ 2∥ f (t,ω)∥. Then by

product measurability, ω → fn (·,ω) is measurable into V ′ because if g ∈ V then

ω → ⟨ fn (·,ω) ,g⟩

is of the form

ω →mn

∑i=1

∫ T

0

⟨cn

i XEni(t,ω) ,g(t)

⟩dt which is ω →

mn

∑i=1

∫ T

0⟨cn

i ,g(t)⟩XEni(t,ω)dt