73.3. PRELIMINARY RESULTS 2465

equals Y in Lq′([0,T ] ;Lq′ (Ω,V ′)

).Thus, by Theorem 34.2.9, for a.e. t, say t /∈ N̂ ⊆

[0,T ] ,m(N̂)= 0,

B(

X (t)−∫ t

0Z (s)dW (s)

)= BX0 +

∫ t

0Y (s)ds in Lq′ (

Ω,V ′).

That is,

(BX)(t) = BX0 +∫ t

0Y (s)ds+B

∫ t

0Z (s)dW (s)

holds in Lq′ (Ω,V ′) where (BX)(t) = B(X (t)) a.e. t, in addition to holding for all t foreach ω . Now let

{tnk

}mn∞

k=1n=1 be partitions for which, from Lemma 73.1.1 there are leftand 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 Lp′ (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)) .

We 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 NC

ω . 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

Thus the conclusion of the above discussion is that at the mesh points, 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. As a case of this, we obtainthe following interesting lemma.