2692 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS
Then also
P(∫ T
0∥un−um∥α
V̂ ds > λ
)≤ e(m,n)
λ
and so there exists a subsequence, denoted by r such that
P(∫ T
0∥ur−ur+1∥α
V̂ ds≤ 2−r)< 2−r
Thus, by the Borel Cantelli lemma, there is a further enlarged set of measure zero, stilldenoted as N such that for ω /∈ N∫ T
0∥ur−ur+1∥α
V̂ ds≤ 2−r
for all r large enough. Hence, by the usual proof of completeness, for these ω,
{ur (·,ω)}
is Cauchy in Lα([0,T ] ,V̂
)and also ur (t,ω) converges to some u(t,ω) pointwise in V̂ for
a.e. t. In addition, from 79.1.20 these functions are a Cauchy sequence in Lα([0,T ]×Ω;V̂
)with respect to the σ algebra of progressively measurable sets. Thus from Lemma 76.3.4,it can be assumed that for ω off the set of measure zero, (t,ω)→ u(t,ω) is progressivelymeasurable. From now on, this will be the sequence or a further subsequence. For ω /∈ N,a set of measure zero and 79.1.18, there is a further subsequence for which the followingconvergences occur as r→ ∞.
ur→ u weakly in V (79.1.21)
B(ur)→ B(u) weakly in V ′ (79.1.22)
zr→ z weakly in V ′ (79.1.23)(B(
ur−∫ (·)
0ΦrdW
))′→(
B(
u−∫ (·)
0ΦdW
))′weakly in V ′ (79.1.24)
∫ (·)
0ΦrdW →
∫ (·)
0ΦdW uniformly in C ([0,T ] ;W ) (79.1.25)
Bur (t)→ Bu(t) weakly in V ′ (79.1.26)
Bu(0) = Bu0, (79.1.27)
Bu(t) = B(u(t)) a.e. t (79.1.28)
In addition to this, we can choose the subsequence such that
supr
supt ̸=s
∥∥∫ ts ΦrdW
∥∥|t− s|γ
<C (ω)< ∞ (79.1.29)
This is thanks to Corollary 79.1.1. The boundedness of the operator A, in particular thegiven estimates, imply that zr is bounded in Lp′ ([0,T ]×Ω,V ′) . Thus a subsequence canbe obtained which yields weak convergence of zr in Lp′ ([0,T ]×Ω,V ′) and then Lemma