Chapter 74

A More Attractive VersionThe following lemma is convenient.

Lemma 74.0.1 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.Also, we have the approximation lemma proved earlier, Lemma 65.3.1.

Lemma 74.0.2 Let Φ : [0,T ]×Ω→V, be B ([0,T ])×F measurable and suppose

Φ ∈ K ≡ Lp ([0,T ]×Ω;E) , p≥ 1

Then there exists a sequence of nested partitions, Pk ⊆Pk+1,

Pk ≡{

tk0 , · · · , tk

mk

}such that the step functions given by

Φrk (t) ≡

mk

∑j=1

Φ

(tk

j

)X(tk

j−1,tkj ](t)

Φlk (t) ≡

mk

∑j=1

Φ

(tk

j−1

)X[tk

j−1,tkj )(t)

both converge to Φ in K as k→ ∞ and

limk→∞

max{∣∣∣tk

j − tkj+1

∣∣∣ : j ∈ {0, · · · ,mk}}= 0.

2497