73.6. CONVERGENCE 2483
However, ∫Ω
∫ T
0∥Z∥2
L2m∥B∥dtdP < ∞
by the assumptions on Z. Therefore, P(A∩ [τm = ∞]) = 0. It follows that
P(A) = ∑m
P(A∩ ([τm = ∞]\ [τm−1 < ∞])) = ∑m
0 = 0
It follows that P(∫ T
0
∣∣(Z ◦ J−1)∗BX ◦ J
∣∣2 dt < ∞
)= 1 and so from Definition 65.10.3, one
can define∫ t
0(Z ◦ J−1
)∗BX ◦ JdW as a local martingale.Convergence will be shown for a subsequence and from now on every sequence will
be a subsequence of this one. As part of Lemma 73.4.2, see 73.4.17, it was shown thatBX ∈ L2 ([0,T ]×Ω,W ′). Therefore, there exist partitions of [0,T ] like the above such that
BX rk ,BX l
k → BX in L2 ([0,T ]×Ω,W ′)
in addition to the convergence of X lk ,X
rk to X in K. From now on, the argument will involve
a subsequence of these.
Lemma 73.6.2 There exists a subsequence still denoted with the subscript k and an en-larged set of measure zero N including the earlier one such that BX l
k (t) ,BX rk (t) also con-
verges pointwise a.e. t to BX (t) in W ′ and X lk (t) ,X
rk (t) converge pointwise a.e. in V to
X (t) for ω /∈ N as well as having convergence of X lk (·,ω) to X (·,ω) in Lp ([0,T ] ;V ) and
BX lk (·,ω) to BX (·,ω) in L2 ([0,T ] ;W ).
Proof: To see that such a sequence exists, let nk be such that∫Ω
∫ T
0
∥∥BX rnk(t)−BX (t)
∥∥2W ′
dtdP+∫
Ω
∫ T
0
∥∥X rnk(t)−X (t)
∥∥pV
dtdP+
∫Ω
∫ T
0
∥∥∥BX lnk(t)−BX (t)
∥∥∥2
W ′dtdP+
∫Ω
∫ T
0
∥∥∥X lnk(t)−X (t)
∥∥∥p
VdtdP < 4−k.
Then
P(∫ T
0
∥∥∥BX lnk(t)−BX (t)
∥∥∥2
W ′dt +
∫ T
0
∥∥X rnk(t)−X (t)
∥∥pV
dt +
∫ T
0
∥∥∥BX lnk(t)−BX (t)
∥∥∥2
W ′dt +
∫ T
0
∥∥∥X lnk(t)−X (t)
∥∥∥p
Vdt > 2−k
)≤ 2k
(4−k)= 2−k
and so by Borel Cantelli lemma, there is a set of measure zero N such that if ω /∈ N,∫ T
0
∥∥∥BX lnk(t)−BX (t)
∥∥∥2
W ′dt +
∫ T
0
∥∥X rnk(t)−X (t)
∥∥pV
dt+
∫ T
0
∥∥∥BX lnk(t)−BX (t)
∥∥∥2
W ′dt +
∫ T
0
∥∥∥X lnk(t)−X (t)
∥∥∥p
Vdt ≤ 2−k