678 CHAPTER 24. THE BOCHNER INTEGRAL

Now∫

Ωx(s)dµ ∈ X = L1 (B,ν) so it makes sense to ask for (

∫Ω

x(s)dµ)(t), at least µ

a.e. t. To find what this is, note∥∥∥∥∫Ω

xn (s)dµ−∫

Ω

x(s)dµ

∥∥∥∥X≤∫

Ω

∥xn (s)− x(s)∥X dµ.

Therefore, since the right side converges to 0,

limn→∞

∥∥∥∥∫Ω

xn (s)dµ−∫

Ω

x(s)dµ

∥∥∥∥X=

limn→∞

∫B

∣∣∣∣(∫Ω

xn (s)dµ

)(t)−

(∫Ω

x(s)dµ

)(t)∣∣∣∣dν = 0.

But (∫Ω

xn (s)dµ

)(t) =

∫Ω

xn (s, t)dµ a.e. t.

Therefore

limn→∞

∫B

∣∣∣∣∫Ω

xn (s, t)dµ−(∫

Ω

x(s)dµ

)(t)∣∣∣∣dν = 0. (24.28)

Also, since xn→ y in L1 (Ω×B),

0 = limn→∞

∫B

∫Ω

|xn (s, t)− y(s, t)|dµdν ≥

limn→∞

∫B

∣∣∣∣∫Ω

xn (s, t)dµ−∫

Ω

y(s, t)dµ

∣∣∣∣dν . (24.29)

From 24.28 and 24.29 ∫Ω

y(s, t)dµ =

(∫Ω

x(s)dµ

)(t) a.e. t.

Thus the following theorem is obtained.

Theorem 24.6.1 Let X = L1 (B) where (B,F ,ν) is a σ finite measure space andlet x ∈ L1 (Ω;X). Then there exists a measurable representative, y ∈ L1 (Ω×B), such that

x(s) = y(s, ·) a.e. s in Ω, the equation in L1 (B) ,

and ∫Ω

y(s, t)dµ =

(∫Ω

x(s)dµ

)(t) a.e. t.

24.7 Vector MeasuresThere is also a concept of vector measures.

Definition 24.7.1 Let (Ω,S ) be a set and a σ algebra of subsets of Ω. A mapping

F : S → X