2694 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS
+
(∫Ω
∥Φn∥2L∞([0,T ],L2)
)1/2(∫Ω
∫ T
0⟨Bur−Bu,ur−u⟩dtdP
)1/2
(79.1.30)
Letting the ei be the special vectors of Theorem 77.2.19,∫Ω
∫ T
0⟨Bur−Bu,ur−u⟩dtdP =
∫Ω
∫ T
0∑
i⟨Bur−Bu,ei⟩2 dtdP
=∫
Ω
∫ T
0∑
ilim inf
p→∞
⟨Bur−Bup,ei
⟩2 dtdP
≤ lim infp→∞
∫Ω
∫ T
0∑
i
⟨Bur−Bup,ei
⟩2 dtdP
= lim infp→∞
∫Ω
∫ T
0∑
i
⟨Bur−Bup,ei
⟩2 dtdP
= lim infp→∞
∫Ω
∫ T
0
⟨Bur−Bup,ur−up
⟩dtdP
Now by 79.1.17, the last expression is no larger than T/2r and so∫Ω
∫ T
0⟨Bur−Bu,ur−u⟩dtdP≤ T
2r
Then, from 79.1.30,
E(∣∣∣∣∫ T
0
(Φr ◦ J−1)∗Bur ◦ JdW −
∫ T
0
(Φ◦ J−1)∗Bu◦ JdW
∣∣∣∣)
≤(∫
Ω
supt⟨Bur (t) ,ur (t)⟩dP
)1/2(∫Ω
∫ T
0∥Φr−Φ∥2 dt
)1/2
+C(
T2r
)1/2
≤C(∫
Ω
∫ T
0∥Φr−Φ∥2 dt
)1/2
+C(
T2r
)1/2
<C2−r +C(
T2r
)1/2
which clearly converges to 0 as r→ ∞. Since the right side is summable, one obtains alsopointwise convergence. This proves the claim.
From the above considerations using the space V̂ , it follows that this u is the same asthe one just obtained in the sense that for ω off N, the two are equal for a.e. t. Thuswe take u to be this common function. Hence there is a set of measure zero such that(t,ω)→XNC u(t,ω) is progressively measurable in the above convergences. Also, thisshows that we are taking u ∈ Lp ([0,T ]×Ω;V ). From the measurability of ur, u, we canobtain a dense countable subset {tk} and an enlarged set of measure zero N such that forω /∈ N,Bu(tk,ω) = B(u(tk,ω)) and Bur (tk,ω) = B(ur (tk,ω)) for all tk and r. This usesthe same argument as in Lemma 73.3.1.
It remains to verify that z(·,ω) ∈ A(u(·,ω) ,ω). It follows from the above consid-erations that the Ito formula above can be used at will. Assume that for a given ω /∈