634 CHAPTER 20. REPRESENTATION THEOREMS

The following lemma says that the difference of regular complex measures is also reg-ular.

Lemma 20.8.2 Suppose λ i, i = 1,2 is a complex Borel measure with total variation finite2

defined on X , a locally compact Hausdorf space. Then λ 1−λ 2 is also a regular measureon the Borel sets.

Proof: Let E be a Borel set. That way it is in the σ algebras associated with bothλ i. Then by regularity of λ i, there exist K and V compact and open respectively such thatK ⊆ E ⊆V and |λ i|(V \K)< ε/2. Therefore,

∑A∈π(V\K)

|(λ 1−λ 2)(A)| = ∑A∈π(V\K)

|λ 1 (A)−λ 2 (A)|

≤ ∑A∈π(V\K)

|λ 1 (A)|+ |λ 2 (A)|

≤ |λ 1|(V \K)+ |λ 2|(V \K)< ε.

Therefore, |λ 1−λ 2|(V \K)≤ ε and this shows λ 1−λ 2 is regular as claimed.

Theorem 20.8.3 Let L ∈C0 (X)′ Then there exists a unique complex measure, λ with |λ |regular and Borel, such that for all f ∈C0 (X) ,

L( f ) =∫

Xf dλ .

Furthermore, ||L||= |λ |(X) .

Proof: By Corollary 20.7.1 there exists σ ∈ L∞ (X ,µ) where µ is a Radon measuresuch that for all f ∈C0 (X) ,

L( f ) =∫

Xf σdµ.

Let a complex Borel measure, λ be given by

λ (E)≡∫

Eσdµ.

This is a well defined complex measure because µ is a finite measure. By Corollary 20.2.10

|λ |(E) =∫

E|σ |dµ (20.8.21)

and σ = g |σ | where gd |λ | is the polar decomposition for λ . Therefore, for f ∈C0 (X) ,

L( f ) =∫

Xf σdµ =

∫X

f g |σ |dµ =∫

Xf gd |λ | ≡

∫X

f dλ . (20.8.22)

From 20.8.21 and the regularity of µ, it follows that |λ | is also regular.

2Recall this is automatic for a complex measure.