2714 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS
Lemma 79.4.7 There exists a set M ⊆ [0,T ] having measure zero such that for t /∈M,
lim infn→∞
∫Ω
|un (t,ω)|2 dP≥∫
Ω
|u(t,ω)|2H dP
We will always assume T /∈M because otherwise, we could complete the argument withT̂ /∈M arbitrarily close to T and then draw the desired conclusions or settle for drawing thedesired conclusions with
[0, T̂]
replacing [0,T ] where T̂ is a fixed number as close to T asdesired. Note that the inequality is only strengthened by going to a subsequence.
Then from the Ito formula and 79.4.44,
12|un (T )|2H −
12|u0|2H +
1n
∫ T
0⟨Fun,un⟩ds+
∫ T
0⟨zn,un⟩ds
−12
∫ T
0||Φ||2L2
ds =∫ T
0⟨ f ,un⟩ds+Mn (T ) (79.4.55)
where Mn (t) is a local martingale, M (0) = 0. Therefore,∫Ω
∫ T
0⟨zn,un⟩dsdP≤
∫Ω
∫ T
0⟨ f ,un⟩dsdP− 1
2
∫Ω
(|un (T )|2H
)dP+
12
∫Ω
(|u0|2
)dP
+12
∫Ω
∫ T
0||Φ||2L2
dsdP+
=0︷ ︸︸ ︷∫Ω
Mn (T )dP
To make more precise, one would use a localizing sequence of stopping times for the localmartingale, take expectations and then pass to a limit, but the end result will be as above.Then taking limsupn→∞ of both sides and using Lemma 79.4.7,
lim supn→∞
∫Ω
∫ T
0⟨zn,un⟩dsdP
≤∫
Ω
∫ T
0⟨ f ,u⟩dsdP− 1
2lim inf
n→∞
∫Ω
|un (T )|2H dP
+12
∫Ω
(|u0|2
)dP+
12
∫Ω
∫ T
0||Φ||2L2
dsdP
≤∫
Ω
∫ T
0⟨ f ,u⟩dsdP+
12
∫Ω
(|u0|2
)dP
+12
∫Ω
∫ T
0||Φ||2L2
dsdP− 12
∫Ω
|u(T )|2H dP
On the other hand, from 79.4.48 and the Ito formula,
⟨z,u⟩V ′,V =∫
Ω
∫ T
0⟨ f ,u⟩dsdP+
12
∫Ω
(|u0|2
)dP
+12
∫Ω
∫ T
0||Φ||2L2
dsdP− 12
∫Ω
|u(T )|2H dP