686 CHAPTER 24. THE BOCHNER INTEGRAL

Letting h be a simple function in Lp (Gn;X),

j∗l (h) = l ( jh) =∫

Gn

g(s)(h(s))dµ. (24.39)

Since the simple functions are dense in Lp (Gn;X), and g ∈ Lp′ (Gn;X ′), it follows 24.39holds for all h ∈ Lp (Gn;X). By Theorem 24.8.2,

∥g∥Lp′ (Gn;X ′) = ∥ j∗l∥(Lp(Gn;X))′ ≤ ∥l∥(Lp(Ω;X))′ .

By the monotone convergence theorem, ∥g∥Lp′ (Ω;X ′) = limn→∞ ∥g∥Lp′ (Gn;X ′)≤∥l∥(Lp(Ω;X))′ .

Therefore g ∈ Lp′ (Ω;X ′) and since simple functions are dense in Lp (Ω;X), 24.38 holdsfor all h ∈ Lp (Ω;X) . Thus l = θg and the theorem is proved because, by Theorem 24.8.2,∥l∥= ∥g∥ and so the mapping θ is onto because l was arbitrary. ■

As in the scalar case, everything generalizes to the case of σ finite measure spaces. Theproof is almost identical.

Lemma 24.8.5 Let (Ω,S ,µ) be a σ finite measure space and let X be a Banach spacesuch that X ′ has the Radon Nikodym property. Then there exists a measurable function, rsuch that r (x)> 0 for all x, such that |r (x)|< M for all x, and

∫rdµ < ∞. For

Λ ∈ (Lp(Ω;X))′, p≥ 1,

there exists a unique h ∈ Lp′(Ω;X ′), L∞(Ω;X ′) if p = 1 such that Λ f =∫

h( f )dµ. Also∥h∥= ∥Λ∥. (∥h∥= ∥h∥p′ if p > 1, ∥h∥∞ if p = 1). Here 1

p +1p′ = 1.

Proof: First suppose r exists as described. Also, to save on notation and to emphasizethe similarity with the scalar case, denote the norm in the various spaces by |·|. Define anew measure µ̃ , according to the rule

µ̃ (E)≡∫

Erdµ. (24.40)

Thus µ̃ is a finite measure on S . Now define a mapping, η : Lp(Ω;X ,µ)→ Lp(Ω;X , µ̃)

by η f = r−1p f . Then

∥η f∥pLp(µ̃)

=∫ ∣∣∣r− 1

p f∣∣∣p rdµ = ∥ f∥p

Lp(µ)

and so η is one to one and in fact preserves norms. I claim that also η is onto. To see this,let g ∈ Lp(Ω;X , µ̃) and consider the function, r

1p g. Then∫ ∣∣∣r 1

p g∣∣∣p dµ =

∫|g|p rdµ =

∫|g|p dµ̃ < ∞

Thus r1p g ∈ Lp (Ω;X ,µ) and η

(r

1p g)= g showing that η is onto as claimed. Thus η is

one to one, onto, and preserves norms. Consider the diagram below which is descriptive ofthe situation in which η∗ must be one to one and onto.

h,Lp′ (µ̃) Lp (µ̃)′ , Λ̃

η∗

→ Lp (µ)′ ,Λ

Lp (µ̃)η

← Lp (µ)