2236 CHAPTER 65. STOCHASTIC INTEGRATION

Thus Pnk consists of points of [0,T ] which are of this form and these partitions are nested.

Define Φlk (0) ≡ 0, Φl

k (t) ≡ Φ

((γnk

(t− s)+ s)+)

. Now suppose N1 is a set of measure

zero. Can s be chosen such that all jumps for all partitions occur off N1? Let (a,b) bean interval contained in [0,T ]. Let S j be the points of (a,b) which are translations of themeasure zero set N1 by t l

j for some j. Thus S j has measure 0. Now pick s ∈ (a,b)\∪ jS j.It will be assumed that all these mesh points miss the set of all t such that ω→Φ(t,ω)

is not in Lp (Ω;E). To get the other sequence of step functions, the right step functions, justuse a similar argument with δ n in place of γn. Just apply the argument to a subsequence ofnk so that the same s can hold for both.

The following proposition says that elementary functions can be used to approximateprogressively measurable functions under certain conditions.

Proposition 65.3.2 Let Φ ∈ Lp ([0,T ]×Ω,E) , p≥ 1, be progressively measurable. Thenthere exists a sequence of elementary functions which converges to Φ in

Lp ([0,T ]×Ω,E) .

These elementary functions have values in E0, a dense subset of E. If εn→ 0, and

Φn (t) =mn

∑k=1

ΨnkX(tk,tk+1] (t)

Ψnk having values in E0, it can be assumed that

mn

∑k=1||Ψn

k−Φ(tk)||Lp(Ω;E) < εn. (65.3.4)

Proof: By Lemma 65.3.1 there exists a sequence of step functions

Φlk (t) =

mk

∑j=1

Φ

(tk

j−1

)X(tk

j−1,tkj ](t)

which converges to Φ in Lp ([0,T ]×Ω,E) where at the left endpoint Φ(0) can be modifiedas described above. Now each Φ

(tk

j−1

)is in Lp (Ω,E) and is F

(tk

j−1

)measurable and so

it can be approximated as closely as desired in Lp (Ω) with a simple function

s(

tkj−1

)≡

mk

∑i=1

c ji XFi (ω) , Fi ∈F

(tk

j−1

).

Furthermore, by density of E0 in E, it can be assumed each c ji ∈ E0 and the condition 65.3.4

holds. Replacing each Φ

(tk

j−1

)with s

(tk

j−1

), the result is an elementary function which

approximates Φlk.

Of course everything in the above holds with obvious modifications replacing [0,T ]with [a,T ] where a < T .

Here is another interesting proposition about the time integral being adapted.