77.2. SOME FUNDAMENTAL THEOREMS 2609

for some C (t,ω) . Here m does not depend on φ . Thus, in particular, this holds for asubsequence and so for each t /∈ N (ω) ,(t,ω) ∈ G because for each φ ∈D ,

limmφ→∞

mφ

∫ t

lmφ(t)⟨φ ,v(s,ω)⟩ds exists and satisfies the above inequality.

Hence, for all ψ ∈M,

Λ(t,ω)ψ = ⟨ψ,u(t,ω)⟩V ′,V ,

where u is product measurable.Also, for t /∈ N (ω) and φ ∈D ,

⟨φ ,u(t,ω)⟩V ′,V = Λ(t,ω)φ ≡ limmφ→∞

mφ

∫ t

lmφ(t)⟨φ ,v(s,ω)⟩ds = ⟨φ ,v(t,ω)⟩V ′,V

therefore, for all φ ∈M⟨φ ,u(t,ω)⟩V ′,V = ⟨φ ,v(t,ω)⟩V ′,V

and hence u(t,ω) = v(t,ω). Thus, for each ω, the product measurable function u satisfiesu(t,ω) = v(t,ω) for a.e. t. Hence u(·,ω) = v(·,ω) in V .

Of course a similar theorem will hold with essentially the identical proof if the functionstake values in V ′.

One can also combine the two theorems to obtain a useful result for limits of functionsin V ′ and V . You just let

X =∞

∏k=1

C ([0,T ])×C ([0,T ])

and let {φ k} be a dense subset of V ′ while {ηk} is a dense subset of V . Then the mappingsare given by

ψm,φ u(t) = m∫ t

lm(t)⟨φ ,u(s)⟩V ′,V ds, ψm,η y(t) = m

∫ t

lm(t)⟨η ,y(s)⟩V,V ′ ds

and one considers for each φ k,ηk,(mk

∫ t

lmk (t)⟨φ k,u(s)⟩V ′,V ds, mk

∫ t

lmk (t)⟨ηk,y(s)⟩V,V ′ ds

)

This time you need to use an enumeration of N×V ′×V and in the last step, you must usea subsequence still denoted with k such that mk → ∞ but φ k = φ and ηk = η for φ ,η twogiven elements of V ′ and V respectively. Then repeating the above argument, one obtainsthe following generalization.

Theorem 77.2.17 Let V be a reflexive separable Banach space with dual V ′, and let p, p′

be such that p > 1 and 1p + 1

p′ = 1. Let the functions t → un (t,ω), for n ∈ N, be inLp ([0,T ] ;V )≡ V and (t,ω)→ un (t,ω) be B ([0,T ])×F ≡P measurable into V . Also