65.3. APPROXIMATING WITH ELEMENTARY FUNCTIONS 2235

=∫

Ω

∫R

∫ T

0||Φ(γn (t− s)+ s)−Φ(t)||pE dtdsdP

+∫

Ω

∫R

∫ 2T+s

−2T+sX

[0,T ]C ||Φ(γn (t− s)+ s)−Φ(t)||pE dtdsdP

Also by definition, γn (t− s)+ s is within 2−n of t and so the integrand in the integral onthe right equals 0 unless t ∈ [−2−n−T,T +2−n]⊆ [−2T,2T ]. Thus the above reduces to∫

Ω

∫R

∫ 2T

−2T||Φ(γn (t− s)+ s)−Φ(t)||pE dtdsdP.

Now Fubini again. ∫R

∫Ω

∫ 2T

−2T||Φ(γn (t− s)+ s)−Φ(t)||pE dtdPds

This converges to 0 as n→ ∞ as was shown above. Therefore,∫ T

0

∫Ω

∫ T

0||Φ(γn (t− s)+ s)−Φ(t)||pE dtdPds

also converges to 0 as n→ ∞. The only problem is that γn (t− s) + s ≥ t − 2−n and soγn (t− s)+ s could be less than 0 for t ∈ [0,2−n]. Since this is an interval whose measureconverges to 0 it follows∫ T

0

∫Ω

∫ T

0

∣∣∣∣Φ((γn (t− s)+ s)+)−Φ(t)

∣∣∣∣pE dtdPds

converges to 0 as n→ ∞. Let

mn (s) =∫

Ω

∫ T

0

∣∣∣∣Φ((γn (t− s)+ s)+)−Φ(t)

∣∣∣∣pE dtdP

Then letting µ denote Lebesgue measure,

µ ([mn (s)> λ ])≤ 1λ

∫ T

0mn (s)ds.

It follows there exists a subsequence nk such that

µ

([mnk (s)>

1k

])< 2−k

Hence by the Borel Cantelli lemma, there exists a set of measure zero N such that for s /∈N,

mnk (s)≤ 1/k

for all k sufficiently large. Pick such an s. Then consider t→Φ

((γnk

(t− s)+ s)+)

. For

nk, t→(

γnk(t− s)+ s

)+has jumps at points of the form 0, s+ l2−nk where l is an integer.