632 CHAPTER 20. REPRESENTATION THEOREMS

Now if E is measurable, the regularity of µ implies there exists a sequence of boundedfunctions fn ∈Cc (X) such that fn (x)→XE (x) a.e. Then using the dominated convergencetheorem in the above,∫

Edµ = lim

n→∞

∫X

fndµ ≥ limn→∞

∣∣∣∣∫Xfnσdµ

∣∣∣∣= ∣∣∣∣∫Eσdµ

∣∣∣∣and so if µ (E)> 0,

1≥∣∣∣∣ 1µ (E)

∫E

σdµ

∣∣∣∣which shows from Lemma 20.2.7 that |σ | ≤ 1 a.e. But also, choosing f1 appropriately,|| f1||∞ ≤ 1, and letting ωL f1 = |L f1| ,

µ (X) = ||L||= sup|| f ||∞≤1

|L f | ≤ |L f1|+ ε

≤∫

Xf1ωσdµ + ε =

∫X

Re( f1ωσ)dµ + ε

≤∫

X|σ |dµ + ε

and since ε is arbitrary,

µ (X)≤∫

X|σ |dµ

which requires |σ | = 1 a.e. since it was shown to be no larger than 1 and if it is smallerthan 1 on a set of positive measure, then the above could not hold.

It only remains to verify µ = |ν |. By Corollary 20.2.10,

|ν |(E) =∫

E|σ |dµ =

∫E

1dµ = µ (E)

and so µ = |ν | . This proves the Theorem.Sometimes people write ∫

Xf dν ≡

∫X

f σd |ν |

where σd |ν | is the polar decomposition of the complex measure ν . Then with this conven-tion, the above representation is

L( f ) =∫

Xf dν , |ν |(X) = ||L|| .

20.7 The Dual Space Of C0(X), Another ApproachIt is possible to obtain the above theorem by a slick trick after first proving it for the specialcase where X is a compact Hausdorff space. For X a locally compact Hausdorff space, X̃denotes the one point compactification of X . Thus, X̃ = X ∪{∞} and the topology of X̃consists of the usual topology of X along with all complements of compact sets which aredefined as the open sets containing ∞. Also C0 (X) will denote the space of continuousfunctions, f , defined on X such that in the topology of X̃ , limx→∞ f (x) = 0. For this spaceof functions, || f ||0 ≡ sup{| f (x)| : x ∈ X} is a norm which makes this into a Banach space.Then the generalization is the following corollary.