674 CHAPTER 24. THE BOCHNER INTEGRAL

Proof: Let ε > 0 be given. Then by uniform integrability there exists δ > 0 such thatif µ (E)< δ then ∫

E∥ fn∥dµ < ε/3.

By Fatou’s lemma the same inequality holds for f . Fatou’s lemma shows f ∈ L1 (Ω;X), fbeing measurable because of Theorem 9.1.2.

By Egoroff’s theorem, Theorem 24.1.18, there exists a set of measure less than δ , Esuch that the convergence of { fn} to f is uniform off E. Therefore,∫

Ω

∥ f − fn∥dµ ≤∫

E(∥ f∥X +∥ fn∥X )dµ +

∫EC∥ f − fn∥X dµ

<2ε

3+∫

EC

ε

(µ (Ω)+1)3dµ < ε

if n is large enough. ■Note that a convenient way to achieve uniform integrability is to say { fn} is bounded

in Lp (Ω;X) for some p > 1. This follows from Holder’s inequality.∫E∥ fn∥dµ ≤

(∫E

dµ

)1/p′(∫Ω

∥ fn∥p dµ

)1/p

≤Cµ (E)1/p′ .

The following theorem is interesting.

Theorem 24.5.8 Let 1 ≤ p < ∞ and let p < r ≤ ∞. Then Lr ([0,T ] ,X) is a Borelsubset of Lp ([0,T ] ;X). Letting C ([0,T ] ;X) denote the continuous functions having valuesin X, C ([0,T ] ;X) is also a Borel subset of Lp ([0,T ] ;X) . Here the measure is ordinary onedimensional Lebesgue measure on [0,T ].

Proof: First consider the claim about Lr ([0,T ] ;X). Let

BM ≡{

x ∈ Lp ([0,T ] ;X) : ∥x∥Lr([0,T ];X) ≤M}.

Then BM is a closed subset of Lp ([0,T ] ;X) . Here is why. If {xn} is a sequence ofelements of BM and xn → x in Lp ([0,T ] ;X) , then passing to a subsequence, still denotedby xn, it can be assumed xn (s)→ x(s) a.e. Hence Fatou’s lemma can be applied to conclude∫ T

0∥x(s)∥r ds≤ lim inf

n→∞

∫ T

0∥xn (s)∥r ds≤Mr < ∞.

Now ∪∞M=1BM = Lr ([0,T ] ;X) . Note this did not depend on the measure space used. It

would have been equally valid on any measure space.Consider now C ([0,T ] ;X) . The norm on this space is the usual norm, ∥·∥

∞. The argu-

ment above shows ∥·∥∞

is a Borel measurable function on Lp ([0,T ] ;X) . This is becauseBM ≡{x ∈ Lp ([0,T ] ;X) : ∥x∥

∞≤M} is a closed, hence Borel subset of Lp ([0,T ] ;X). Now

let θ ∈ L (Lp ([0,T ] ;X) ,Lp (R;X)) such that θ (x(t)) = x(t) for all t ∈ [0,T ] and alsoθ ∈ L (C ([0,T ] ;X) ,BC (R;X)) where BC (R;X) denotes the bounded continuous func-tions with a norm given by ∥x∥ ≡ supt∈R ∥x(t)∥ , and θx has compact support.

For example, you could define

x̃(t)≡

x(t) if t ∈ [0,T ]x(2T − t) if t ∈ [T,2T ]x(−t) if t ∈ [−T,0]0 if t /∈ [−T,2T ]