20.8. MORE ATTRACTIVE FORMULATIONS 633

Corollary 20.7.1 Let L ∈ (C0 (X))′ where X is a locally compact Hausdorff space. Thenthere exists σ ∈ L∞ (X ,µ) for µ a finite Radon measure such that for all f ∈C0 (X),

L( f ) =∫

Xf σdµ.

Proof: LetD̃≡

{f ∈C

(X̃)

: f (∞) = 0}.

Thus D̃ is a closed subspace of the Banach space C(

X̃)

. Let θ : C0 (X)→ D̃ be defined by

θ f (x) ={

f (x) if x ∈ X ,0 if x = ∞.

Then θ is an isometry of C0 (X) and D̃. (||θu||= ||u|| .)The following diagram is obtained.

C0 (X)′θ∗←

(D̃)′ i∗← C

(X̃)′

C0 (X) →θ

D̃ →i

C(

X̃)

By the Hahn Banach theorem, there exists L1 ∈C(

X̃)′

such that θ∗i∗L1 = L. Now apply

Theorem 20.6.5 to get the existence of a finite Radon measure, µ1, on X̃ and a function

σ ∈ L∞

(X̃ ,µ1

), such that

L1g =∫

X̃gσdµ1.

Letting the σ algebra of µ1 measurable sets be denoted by S1, define

S ≡{E \{∞} : E ∈S1}

and let µ be the restriction of µ1 to S . If f ∈C0 (X),

L f = θ∗i∗L1 f ≡ L1iθ f = L1θ f =

∫X̃

θ f σdµ1 =∫

Xf σdµ.

This proves the corollary.

20.8 More Attractive FormulationsIn this section, Corollary 20.7.1 will be refined and placed in an arguably more attractiveform. The measures involved will always be complex Borel measures defined on a σ

algebra of subsets of X , a locally compact Hausdorff space.

Definition 20.8.1 Let λ be a complex measure. Then∫

f dλ ≡∫

f hd |λ | where hd |λ | isthe polar decomposition of λ described above. The complex measure, λ is called regularif |λ | is regular.