65.2. DIFFERENT DEFINITION OF ELEMENTARY FUNCTIONS 2233

65.2 Different Definition Of Elementary FunctionsWhat if elementary functions had been defined in terms of X[tk,tk+1)? That is, what if theelementary functions had been of the form

Φ(t) =n−1

∑k=0

Φ(tk)X[tk,tk+1) (t)?

Would anything change? If you go over the arguments given, it is clear that nothing wouldchange at all. Furthermore, this elementary function equals the one described above off afinite set of mesh points so the convergence properties in L2

([0,T ]×Ω,L2

(Q1/2U,H

)),

which will be important in what follows are exactly the same. Thus it does not matterwhether we give elementary functions in this form or in the form described above. How-ever, some arguments given later about localization depend on it being in the earlier form.

65.3 Approximating With Elementary FunctionsHere is a really surprising result about approximating with step functions which is dueto Doob. See [78] which is where I found this lemma. This is based on continuity oftranslation in the Lp (R;E) .

Lemma 65.3.1 Let Φ : [0,T ]×Ω→ E, 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.

Also, each Φ

(tk

j

),Φ(

tkj−1

)is in Lp (Ω;E). One can also assume that Φ(0) = 0. The mesh

points{

tkj

}mk

j=0can be chosen to miss a given set of measure zero. In addition to this, we

can assume that ∣∣∣tkj − tk

j−1

∣∣∣= 2−nk

except for the case where j = 1 or j =mnk when this is so, you could have∣∣∣tk

j − tkj−1

∣∣∣< 2−nk .