676 CHAPTER 21. THE BOCHNER INTEGRAL

21.6 Measurable RepresentativesIn this section consider the special case where X = L1 (B,ν) where (B,F ,ν) is a σ finitemeasure space and x ∈ L1 (Ω;X). Thus for each s ∈ Ω, x(s) ∈ L1 (B,ν). In general, themap

(s, t)→ x(s)(t)

will not be product measurable, but one can obtain a measurable representative. This isimportant because it allows the use of Fubini’s theorem on the measurable representative.

By Theorem 21.2.4, there exists a sequence of simple functions, {xn}, of the form

xn (s) =m

∑k=1

akXEk (s) (21.6.25)

where ak ∈ L1 (B,ν) which satisfy the conditions of Definition 21.2.3 and

∥xn− xm∥L1(Ω,L1(B))→ 0 as m,n→ ∞ (21.6.26)

For such a simple function, you can assume the Ek are disjoint and then

∥xn∥L1(Ω,L1(B)) =m

∑k=1∥ak∥L1(B) µ (Ek) =

m

∑k=1

∫B|ak|dνµ (Ek)

=∫

Ω

∫B|ak (t)|dν (t)XEk (s)dµ (s)

=∫

Ω

∫B|xn|dνdµ

Also, each xn is product measurable. Thus from 21.6.26,

∥xn− xm∥L1(Ω,L1(B)) =∫

Ω

∫B|xn− xm|dνdµ

which shows that {xn} is a Cauchy sequence in L1 (Ω×B,µ×λ ) . Then there exists y ∈L1 (Ω×B,µ×λ ) and a subsequence still called {xn} such that

limn→∞

∫Ω

∫B|xn− y|dνdµ = lim

n→∞

∫Ω

∥xn− y∥L1(B) dµ = ∥xn− y∥L1(Ω,L1(B)) = 0.

Now consider 21.6.26. Since limm→∞ xm (s) = x(s) in L1 (B) , it follows from Fatou’slemma that

∥xn− x∥L1(Ω,L1(B)) ≤ lim infm→∞∥xn− xm∥L1(Ω,L1(B)) < ε

for all n large enough. Hence

limn→∞∥xn− x∥L1(Ω,L1(B)) = 0

and sox(s) = y(s) in L1 (B) µ a.e. s