676 CHAPTER 24. THE BOCHNER INTEGRAL

Then

∥ψkxn (t)− xn (t)− (ψkx(t)− x(t))∥

≤ ∥xn (t)− x(t)∥X +

∥∥∥∥∫ 1/k

−1/kφ k (s)(θxn (t− s)−θx(t− s))ds

∥∥∥∥≤ ∥xn (t)− x(t)∥X +Ck,θ ∥xn− x∥Lp(0,T ;X)

which converges to 0 as n→ ∞. It follows that for a.e. t,

∥ψkx(t)− x(t)∥ ≤ α.

Thus S is closed and so the set in 24.25 is a Borel set. ■As in the scalar case, the following lemma holds in this more general context.

Lemma 24.5.9 Let (Ω,µ) be a regular measure space where Ω is a locally compactHausdorff space or more simply a metric space with closed balls compact. Then Cc (Ω;X)the space of continuous functions having compact support and values in X is dense inLp (0,T ;X) for all p ∈ [0,∞). For any measure space (Ω,F ,µ), the simple functions aredense in Lp (0,T ;X) .

Proof: First, the simple functions are dense in Lp (0,T ;X) . Let f ∈ Lp (0,T ;X) and let{xn} denote a sequence of simple functions which converge to f pointwise which also havethe property that

∥xn (s)∥ ≤ 2∥ f (s)∥

Then ∫Ω

∥xn (s)− f (s)∥p dµ → 0

from the dominated convergence theorem. Therefore, the simple functions are indeed densein Lp (0,T ;X) .

Next suppose (Ω,µ) is a regular measure space. If x(s) ≡ ∑i aiXEi (s) is a simplefunction, then by regularity, there exist compact sets Ki and open sets, Vi such that Ki ⊆Ei ⊆Vi and µ (Vi \Ki)

1/p < ε/∑i ||ai|| . Let Ki ≺ hi ≺Vi. Then consider

∑i

aihi ∈Cc (Ω) .

By the triangle inequality,(∫Ω

∥∥∥∥∥∑iaihi (s)−aiXEi (s)

∥∥∥∥∥p

dµ

)1/p

≤ ∑i

(∫Ω

∥ai (hi (s)−XEi (s))∥p dµ

)1/p

≤ ∑i

(∫Ω

∥ai∥p |hi (s)−XEi (s)|p dµ

)1/p

≤∑i∥ai∥

(∫Vi\Ki

dµ

)1/p

≤ ∑i∥ai∥µ (Vi \Ki)

1/p < ε

This and the first part of the lemma shows that Cc (Ω;X) = Lp (Ω;X). ■