684 CHAPTER 21. THE BOCHNER INTEGRAL

Proof: Let l ∈ (Lp (Ω;X))′ and define F (E) ∈ X ′ by

F (E)(x)≡ l (XE (·)x).

Lemma 21.8.4 F defined above is a vector measure with values in X ′ and |F |(Ω)< ∞.

Proof of the lemma: Clearly F (E) is linear. Also

∥F (E)∥= sup∥x∥≤1

∥F (E)(x)∥

≤ ∥l∥ sup∥x∥≤1

∥XE (·)x∥Lp(Ω;X) ≤ ∥l∥µ (E)1/p.

Let {Ei}∞i=1 be a sequence of disjoint elements of S and let E = ∪n<∞En.∣∣∣∣∣F (E)(x)−

n

∑k=1

F (Ek)(x)

∣∣∣∣∣ =

∣∣∣∣∣l (XE (·)x)−n

∑i=1

l (XEi (·)x)

∣∣∣∣∣ (21.8.36)

≤ ∥l∥

∥∥∥∥∥XE (·)x−n

∑i=1

XEi (·)x

∥∥∥∥∥Lp(Ω;X)

≤ ∥l∥µ

(⋃k>n

Ek

)1/p

∥x∥.

Since µ (Ω)< ∞, limn→∞ µ

( ⋃k>n

Ek

)1/p

= 0 and so inequality 21.8.36 shows that

limn→∞

∥∥∥∥∥F (E)−n

∑k=1

F (Ek)

∥∥∥∥∥X ′

= 0.

To show |F |(Ω) < ∞, let ε > 0 be given, let {H1, · · · ,Hn} be a partition of Ω, and let∥xi∥ ≤ 1 be chosen in such a way that

F (Hi)(xi)> ∥F (Hi)∥− ε/n.

Thus

−ε +n

∑i=1∥F (Hi)∥<

n

∑i=1

l (XHi (·)xi)≤ ∥l∥

∥∥∥∥∥ n

∑i=1

XHi (·)xi

∥∥∥∥∥Lp(Ω;X)

≤ ∥l∥(∫

Ω

n

∑i=1

XHi (s)dµ

)1/p

= ∥l∥µ (Ω)1/p.

Since ε > 0 was arbitrary, ∑ni=1 ∥F (Hi)∥ < ∥l∥µ (Ω)1/p.Since the partition was arbitrary,

this shows |F |(Ω)≤ ∥l∥µ (Ω)1/p and this proves the lemma.