77.3. PRELIMINARY RESULTS 2611

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. It is Lemma 34.4.2.

Theorem 77.2.19 Suppose V,W are separable Banach spaces, such that V is dense in Wand B ∈L (W,W ′) satisfies

⟨Bx,x⟩ ≥ 0, ⟨Bx,y⟩= ⟨By,x⟩ ,B ̸= 0.

Then there exists a countable set {ei} of vectors in V such that⟨Bei,e j

⟩= δ i j

and for each x ∈W,

⟨Bx,x⟩=∞

∑i=1|⟨Bx,ei⟩|2 ,

and also

Bx =∞

∑i=1⟨Bx,ei⟩Bei,

the series converging in W ′. In case B = B(ω) where ω → B(ω) is measurable intoL (W,W ′) , these vectors ei will also depend on ω and will be measurable functions ofω .

77.3 Preliminary ResultsWe use the following well known theorem [91]. It is Theorem 34.7.6.

Theorem 77.3.1 Let E ⊆ F ⊆ G where the injection map is continuous from F to G andcompact from E to F. Let p≥ 1, let q > 1, and define

S≡ {u ∈ Lp ([a,b] ,E) : for some C, ∥u(t)−u(s)∥G ≤C |t− s|1/q

and ||u||Lp([a,b],E) ≤ R}.

Thus S is bounded in Lp ([a,b] ,E) and Holder continuous into G. Then S is precompact inLp ([a,b] ,F). This means that if {un}∞

n=1 ⊆ S, it has a subsequence{

unk

}which converges

in Lp ([a,b] ,F) .

We recall the following theorem which is proved in [99] and earlier, Theorem 25.5.2for what will suffice here.