2462 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE

Lemma 73.1.2 Let fn→ f in Lp ([0,T ]×Ω,E) . Then there exists a subsequence nk and aset of measure zero N such that if ω /∈ N, then

fnk (·,ω)→ f (·,ω)

in Lp ([0,T ] ,E) and for a.e. t.

Proof: We have

P([∥ fn− f∥Lp([0,T ],E) > λ

])≤ 1

λ

∫Ω

∥ fn− f∥Lp([0,T ],E) dP

≤ 1λ∥ fn− f∥Lp([0,T ]×Ω,E)

Hence there exists a subsequence nk such that

P([∥∥ fnk − f

∥∥Lp([0,T ],E) > 2−k

])≤ 2−k

Then by the Borel Cantelli lemma, it follows that there exists a set of measure zero N suchthat for all k large enough and ω /∈ N,∥∥ fnk − f

∥∥Lp([0,T ],E) ≤ 2−k

Now by the usual arguments used in proving completeness, fnk (t)→ f (t) for a.e.t.Because of this lemma, it can also be assumed that for a.e. ω, pointwise convergence

is obtained on [0,T ] as well as convergence in Lp ([0,T ]). This kind of assumption will betacitly made whenever convenient.

Also recall the diagram for the definition of the integral which has values in a Hilbertspace W .

U↓ Q1/2

U1 ⊇ JQ1/2U J←1−1

Q1/2U

Zn ↘ ↓ Z

W

The idea was to get∫ t

0 ZdW where Z ∈ L2([0,T ]×Ω;L2

(Q1/2U,W

)). Here W (t) was a

cylindrical Wiener process. This meant that it was a Q1 Wiener process on U1 for Q1 = JJ∗

and J was a Hilbert Schmidt operator mapping Q1/2U to U1. To get∫ t

0 ZdW, Z ◦ J−1 wasapproximated by a sequence of elementary functions {Zn} having values in L (U1,W ) .Then ∫ t

0ZdW ≡ lim

n→∞

∫ t

0ZndW

and this limit exists in L2 (Ω,W ) and is independent of the choice of U1 and J. In fact, U1can be assumed to be U .