670 CHAPTER 21. THE BOCHNER INTEGRAL

Lemma 21.5.2 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, it followsthat 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).

Thus, if N > M, and s is a point where g(s)< ∞,

∥xN+1 (s)− xM+1 (s)∥X ≤N

∑n=M+1

∥xn+1 (s)− xn (s)∥X

≤∞

∑n=M+1

∥xn+1 (s)− xn (s)∥X

which shows that {xN+1 (s)}∞

N=1 is a Cauchy sequence for each s /∈ E. Now let

x(s)≡{

limN→∞ xN (s) if s /∈ E,0 if s ∈ E.

Theorem 21.1.10 shows that x is strongly measurable. By Fatou’s lemma,∫Ω

∥x(s)− xN (s)∥p dµ ≤ lim infM→∞

∫Ω

∥xM (s)− xN (s)∥p dµ.