24.5. THE SPACES Lp (Ω;X) 671

24.5 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 24.5.1 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

. (24.24)

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 foundin the presentation of scalar valued functions it is clear that Lp (Ω;X) is a normed linearspace with the norm given by 24.24. In fact, Lp (Ω;X) is a Banach space. This is the maincontribution of the next theorem.

Lemma 24.5.2 If xn is a Cauchy sequence in Lp (Ω;X) satisfying

∞

∑n=1∥xn+1− xn∥p < ∞,

then there exists x ∈ Lp (Ω;X) such that xn (s)→ x(s) a.e. and

∥x− xn∥p→ 0.

Proof: Let gN (s)≡ ∑Nn=1 ∥xn+1 (s)− xn (s)∥X . Then by the triangle inequality,(∫

Ω

gN (s)p dµ

)1/p

≤N

∑n=1

(∫Ω

∥xn+1 (s)− xn (s)∥p dµ

)1/p

≤∞

∑n=1∥xn+1− xn∥p < ∞.

Let

g(s) = limN→∞

gN (s) =∞

∑n=1∥xn+1 (s)− xn (s)∥X .

By the monotone convergence theorem,(∫Ω

g(s)p dµ

)1/p

= limN→∞

(∫Ω

gN (s)p dµ

)1/p

< ∞.

Therefore, there exists a measurable set of measure 0 called E, such that for s /∈E, g(s)<∞.Hence, for s /∈ E, limN→∞ xN+1 (s) exists because

xN+1 (s) = xN+1 (s)− x1 (s)+ x1 (s) =N

∑n=1

(xn+1 (s)− xn (s))+ x1 (s).