21.5. THE SPACES Lp (Ω;X) 669

Also, ∫Ω1

∥∥∥∥∫Ω2

f (s1,s2)XNC (s1)dλ

∥∥∥∥dµ

≤∫

Ω1

∫Ω2

∥ f (s1,s2)∥dλdµ

=∫

Ω1×Ω2

∥ f (s1,s2)∥d (µ×λ )< ∞.

Therefore, the function in 21.4.22 is indeed Bochner integrable and so in 21.4.20 the φ canbe taken outside the last integral. Thus,

φ

(∫Ω1×Ω2

f (s1,s2)d (µ×λ )

)=

∫Ω1×Ω2

φ ◦ f (s1,s2)d (µ×λ )

=∫

Ω1

∫Ω2

φ ◦ f (s1,s2)dλdµ

=∫

Ω1

φ

(∫Ω2

f (s1,s2)XNC (s1)dλ

)dµ

= φ

(∫Ω1

∫Ω2

f (s1,s2)XNC (s1)dλdµ

).

Since X ′ separates the points,∫Ω1×Ω2

f (s1,s2)d (µ×λ ) =∫

Ω1

∫Ω2

f (s1,s2)XNC (s1)dλdµ.

The other formula follows from similar reasoning.

21.5 The Spaces Lp (Ω;X)

Recall that x is Bochner when it is strongly measurable and∫

Ω∥x(s)∥dµ < ∞. It is natural

to generalize to∫

Ω∥x(s)∥p dµ < ∞.

Definition 21.5.1 x ∈ Lp (Ω;X) for p ∈ [1,∞) if x is strongly measurable and∫Ω

∥x(s)∥p dµ < ∞

Also

∥x∥Lp(Ω;X) ≡ ∥x∥p ≡(∫

Ω

∥x(s)∥p dµ

)1/p

. (21.5.23)

As in the case of scalar valued functions, two functions in Lp (Ω;X) are consideredequal if they are equal a.e. With this convention, and using the same arguments found inthe presentation of scalar valued functions it is clear that Lp (Ω;X) is a normed linear spacewith the norm given by 21.5.23. In fact, Lp (Ω;X) is a Banach space. This is the maincontribution of the next theorem.