2500 CHAPTER 74. A MORE ATTRACTIVE VERSION

holds in Lq′ (Ω,V ′) where (BX)(t) = B(X (t)) a.e. t in this space, for all t /∈ N̂, a set ofLebesgue measure zero, in addition to holding for all t for each ω . Now let

{tnk

}mn∞

k=1n=1be partitions for which, from Lemma 74.0.2 there are left and right step functions X l

k ,Xrk ,

which converge in Lp ([0,T ]×Ω;V ) to X and such that each{

tnk

}mnk=1 has empty intersection

with the set of measure zero N̂ where, in Lq′ (Ω;V ′) , (BX)(t) ̸= B(X (t)) in Lq′ (Ω;V ′).Thus for tk a generic partition point,

BX (tk) = B(X (tk)) in Lq′ (Ω;V ′

)Hence there is an exceptional set of measure zero,N (tk)⊆Ω such that for

ω /∈ N (tk) ,BX (tk)(ω) = B(X (tk,ω)) .

Define an exceptional set N ⊆ Ω to be the union of all these N (tk) . There are countablymany and so N is also a set of measure zero. Then for ω /∈ N, and tk any mesh pointat all, BX (tk)(ω) = B(X (tk,ω)) . This will be important in what follows. In addition tothis, from the integral equation, for each of these ω /∈ N, BX (t)(ω) = B(X (t,ω)) for allt /∈ Nω ⊆ [0,T ] where Nω is a set of Lebesgue measure zero. Thus the tk from the variouspartitions are always in Nω . By Lemma 69.4.1, there exists a countable set {ei} of vectorsin V such that ⟨

Bei,e j⟩= δ i j

and for each x ∈W,

⟨Bx,x⟩=∞

∑i=0|⟨Bx,ei⟩|2 , Bx =

∞

∑i=1⟨Bx,ei⟩Bei

By this lemma, if B = B(ω) where B is F0 measurable into L (W,W ′) , then the ei arealso F0 measurable into V . Thus the conclusion of the above discussion is that at the meshpoints, it is valid to write

⟨(BX)(tk) ,X (tk)⟩ = ⟨B(X (tk)) ,X (tk)⟩= ∑

i⟨(BX)(tk) ,ei⟩2 = ∑

i⟨B(X (tk)) ,ei⟩2

just as would be the case if (BX)(t) = B(X (t)) for every t. In all which follows, the meshpoints will be like this and an appropriate set of measure zero which may be replaced witha larger set of measure zero finitely many times is being neglected. Obviously, one cantake a subsequence of the sequence of partitions described above without disturbing theabove observations. We will denote these partitions as Pk. Thus we obtain the followinginteresting lemma.

Lemma 74.2.1 In the above situation, there exists a set of measure zero N ⊆ Ω and adense subset of [0,T ] , D such that for ω /∈ N, BX (t,ω) = B(X (t,ω)) for all t ∈ D. This

set D is the union of nested paritions {Pk}={

tkj

}mk∞

j=1,k=1such that the left and right step

functions{

X lk

},{

X rk

}converge to X in Lp ([0,T ]×Ω;V ). There is also a set of Lebesgue