27.7. CONVOLUTION AND SUMS 753

Proof: For each x ∈ K, there exists a ball, B(x,δ x) such that B(x,3δ x) ⊆U . Finitelymany of these balls cover K because K is compact, say {B(xi,δ xi)}

mi=1. Let

0 < δ < min(δ xi : i = 1,2, · · · ,m) .

Now pick any x ∈ K. Then x ∈ B(xi,δ xi) for some xi and so

B(x,δ )⊆ B(xi,2δ xi)⊆U.

Therefore, for any x ∈ K,dist(x,UC

)≥ δ . If x ∈ B(xi,2δ xi) for some xi, it follows that

dist(x,UC

)≥ δ because then B(x,δ ) ⊆ B(xi,3δ xi) ⊆U. If x /∈ B(xi,2δ xi) for any of the

xi, then x /∈ B(y,δ ) for any y ∈ K because all these sets are contained in some B(xi,2δ xi) .Consequently dist(x,K)≥ δ . ■

From this lemma, there is an easy corollary.

Corollary 27.7.2 Suppose K is a compact subset of U, an open set in E a metric space.Then there exists a uniformly continuous function f defined on all of E, having values in[0,1] such that f (x) = 0 if x /∈U and f (x) = 1 if x ∈ K.

Proof: Consider

f (x)≡dist(x,UC

)dist(x,UC)+dist(x,K)

.

Then some algebra yields∣∣ f (x)− f(x′)∣∣≤ 1

δ

(∣∣dist(x,UC)−dist

(x′,UC)∣∣+ ∣∣dist(x,K)−dist

(x′,K

)∣∣)where δ is the constant of Lemma 27.7.1. Now it is a general fact that∣∣dist(x,S)−dist

(x′,S

)∣∣≤ d(x,x′).

See Proposition 3.6.6. Therefore, | f (x)− f (x′)| ≤ 2δ

d (x,x′) and this proves the corollary.■

Now suppose µ is a finite measure defined on the Borel sets of a separable Banachspace E. It was shown above that µ is inner and outer regular. Lemma 26.1.9 on Page 717shows that µ is inner regular in the usual sense with respect to compact sets. This makespossible the following theorem.

Theorem 27.7.3 Let µ be a finite measure on B (E) where E is a separable Ba-nach space and let f ∈ Lp (E; µ) . Then for any ε > 0, there exists a uniformly continuous,bounded g defined on E such that

∥ f −g∥Lp(E) < ε.

Proof: As usual in such situations, it suffices to consider only f ≥ 0. Then by Theorem9.1.6 on Page 239 and an application of the monotone convergence theorem, there exists asimple measurable function,

s(x)≡m

∑k=1

ckXAk (x)

such that || f − s||Lp(E) < ε/2. Now by regularity of µ there exist compact sets, Kk and

open sets, Vk such that 2∑mk=1 |ck|µ (Vk \K)1/p < ε/2 and by Corollary 27.7.2 there exist