78.1. SUMMARY OF THE PROBLEM 2659

Next is what it means to be measurable into V or V ′. Such functions have representa-tives which are product measurable.

Lemma 78.1.5 Let f (·,ω)∈V ′. Then if ω→ f (·,ω) is measurable into V ′, it follows thatfor 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 ′. Therefore,

there is no loss of generality in assuming that

ci (t) =N

∑j=1

dijXFj (t)

where dij ∈ V ′. Hence we can assume each fn is product measurable into B (V ′)×F .

Then by Theorem 78.1.4, there exists f̂ (·,ω) ∈ V ′ such that f̂ is product measurable and asubsequence fn(ω) converging weakly in V ′ to f̂ (·,ω) for each ω . Thus fn(ω) (ω)→ f (ω)

strongly in V ′ and fn(ω) (ω)→ f̂ (ω) weakly in V ′. Therefore, f̂ (ω) = f (ω) in V ′ andso it can be assumed that if f is measurable into V ′ then for each ω, it has a representativef̂ (ω) such that (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

and this is F measurable since Eni is product measurable. Thus, it is measurable into V ′

as desired and

⟨ f (·,ω) ,g⟩= limn→∞⟨ fn (·,ω) ,g⟩ , ω → ⟨ fn (·,ω) ,g⟩ is F measurable.

By the Pettis theorem, ω → ⟨ f (·,ω) ,g⟩ is measurable into V ′. Obviously, the conclusionis the same for these two conditions if V ′ is replaced with V .

The following theorem is also useful. It is really a generalization of the familiar GramSchmidt process. See Lemma 34.4.2 on Page 1179.