79.1. THE CASE OF UNIQUENESS 2693
76.3.4 may be applied to conclude that off a set of measure zero, z is progressively measur-able.
The claim 79.1.26 and 79.1.27 follow from the continuity of the evaluation map definedon X , Theorem 77.2.2. The claim in 79.1.28 follows from 79.1.22 and the convergence79.1.26. To see this, let ψ ∈C∞
c (0,T ) .∫ T
0Bu(t)ψ (t)dt = lim
r→∞
∫ T
0Bur (t)ψ (t)dt
= limr→∞
∫ T
0B(ur (t))ψ (t)dt =
∫ T
0B(u(t))ψ (t)dt
Since this is true for all such ψ, it follows that Bu(t) = B(u(t)) for a.e. t. Passing to a limitin the integral equation yields the following for ω off a set of measure zero,
Bu(t,ω)−Bu0 (ω)+∫ t
0z(s,ω)ds =
∫ t
0f (s,ω)ds+B
∫ t
0ΦndW
In the following claim, assume Φ ∈ Lα(Ω,L∞
([0,T ] ,L2
(Q1/2U,W
))),α > 2
Claim: limr→∞
∫ T0(Φr ◦ J−1
)∗Bur ◦ JdW =∫ T
0(Φ◦ J−1
)∗Bu ◦ JdW off a set of mea-sure zero.
Proof of claim:
E(∣∣∣∣∫ T
0
(Φr ◦ J−1)∗Bur ◦ JdW −
∫ T
0
(Φ◦ J−1)∗Bu◦ JdW
∣∣∣∣)
≤ E(∣∣∣∣∫ T
0
(Φr ◦ J−1)∗Bur ◦ JdW −
∫ T
0
(Φ◦ J−1)∗Bur ◦ JdW
∣∣∣∣)+E(∣∣∣∣∫ T
0
(Φ◦ J−1)∗Bur ◦ JdW −
∫ T
0
(Φ◦ J−1)∗Bu◦ JdW
∣∣∣∣)Then, by the Burkholder Davis Gundy inequality,
≤∫
Ω
(∫ T
0∥Φr−Φ∥2 ⟨Bur,ur⟩
)1/2
dP
+∫
Ω
(∫ T
0∥Φ∥2 ⟨Bur−Bu,ur−u⟩
)1/2
dP
≤∫
Ω
supt⟨Bur (t) ,ur (t)⟩1/2
(∫ T
0∥Φr−Φ∥2 dt
)1/2
dP
+∫
Ω
∥Φn∥L∞([0,T ],L2)
(∫ T
0⟨Bur−Bu,ur−u⟩
)1/2
dP
≤(∫
Ω
supt⟨Bur (t) ,ur (t)⟩dP
)1/2(∫Ω
∫ T
0∥Φr−Φ∥2 dt
)1/2