1172 CHAPTER 34. GELFAND TRIPLES AND RELATED STUFF

Lemma 34.3.1 Let Φ : [0,T ]→ E, be Lebesgue measurable and suppose

Φ ∈ K ≡ Lp ([0,T ] ;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.

In the formulas, define Φ(0) = 0. The mesh points{

tkj

}mk

j=0can be chosen to miss a given

set of measure zero.

Note that it would make no difference in terms of the conclusion of this lemma if youdefined

Φlk (t)≡

mk

∑j=1

Φ

(tk

j−1

)X(tk

j−1,tkj ](t)

because the modified function equals the one given above off a countable subset of [0,T ] ,the union of the mesh points.

Proof: For t ∈ R let γn (t)≡ k/2n,δ n (t)≡ (k+1)/2n, where

t ∈ (k/2n,(k+1)/2n],

and 2−n < T/4. Also suppose Φ is defined to equal 0 on [0,T ]C×Ω. There exists a set ofmeasure zero N such that for ω /∈ N, t → ∥Φ(t,ω)∥ is in Lp (R). Therefore by continuityof translation, as n→ ∞ it follows that for ω /∈ N, and t ∈ [0,T ] ,∫

R||Φ(γn (t)+ s)−Φ(t + s)||pE ds→ 0

The above is dominated by∫R

2p−1 (||Φ(s)||p + ||Φ(s)||p)X[−2T,2T ] (s)ds

=∫ 2T

−2T2p−1 (||Φ(s)||p + ||Φ(s)||p)ds < ∞