8.8. CONSTRUCTING MEASURES FROM FUNCTIONALS 193

respectively such that F ⊆ E ⊆ G and µ (G\F) = 0. In particular, µ is inner and outerregular on F . If

(X ,F̂ , µ̂

)has the same properties, outer regular, and σ finite, and µ = µ̂

on open sets, then if both µ, µ̂ are complete measures, it follows that µ = µ̂ and F = F̂ .

Proof: Since µ is outer regular and µ (Xn) < ∞, there exists an open set Vn ⊇ E ∩Xnsuch that

µ (Vn \ (E ∩Xn)) = µ (Vn)−µ (E ∩Xn)<ε

2n .

Then let V ≡ ∪nVn so that V ⊇ E. Then E = ∪nE ∩Xn and so

µ (V \E)≤ µ (∪n (Vn \ (E ∩Xn)))≤∑n

µ (Vn \ (E ∩Xn))< ∑n

ε

2n = ε

Similarly, there exists Un open such that µ(Un \

(EC ∩Xn

))< ε

2n ,Un ⊇ EC ∩Xn so if U ≡∪nUn,µ

(U \EC

)= µ

(E \UC

)< ε. Now UC is closed and contained in E because U ⊇EC.

Hence, letting ε = 12n , there exist closed sets Cn, and open sets Vn such that Cn ⊆E ⊆Vn and

µ (Vn \Cn)<1

2n−1 . Letting G≡∩nVn,F ≡∪nCn,F ⊆ E ⊆G and µ (G\F)≤ µ (Vn \Cn)<1

2n−1 . Since n is arbitrary, µ (G\F) = 0.Let the disjoint sets Xn work for µ̂ as well as for µ . One can simply take an enumeration

of Xn ∩ X̂m where the X̂m work for µ̂ . Let K consist of the open sets.This is clearly aπ system because finite intersections remain in K . Also µ = µ̂ on K by assumption.Let G be those Borel sets F such that µ (F ∩Xn) = µ̂ (F ∩Xn) . Then G is clearly closedwith respect to complements and countable disjoint intersections so G = B (X) . Takingunions, it follows that µ̂ = µ on the Borel sets. Now by the first part, there is G a Gδ

set and F an Fσ such that µ (G\F) = µ̂ (G\F) = 0 and G ⊇ E ⊇ F for E ∈ F . Thenby completeness of µ̂, it follows that E ∈ F̂ . Thus F ⊆ F̂ . Similarly F̂ ⊆F . Also,µ (E) = µ (G) = µ̂ (G) = µ̂ (E) so µ = µ̂ . ■

8.8 Constructing Measures From FunctionalsHere is a theorem which is the main result on measures and functionals defined on a spaceof continuous functions. The typical situation is of a metric space in which closed balls arecompact like Rp.

Definition 8.8.1 Cc (X) will denote the complex values functions which have com-pact support in some metric space X. This is clearly a linear space. Then a linear functionL : Cc (X)→ C is called “postitive” if whenever f ≥ 0, then L f ≥ 0.

The following theorem is called the Riesz representation theorem for positive linearfunctionals. I will make the way in which it represents something more clear later on. Fornow it will just produce lots of measures. Recall that K ≺ f ≺ V means that f is 1 on thecompact set K, has compact support in the open set V and takes values in [0,1]. Also f ≺Vmeans f has values in [0,1] and has compact support in the open set V .

Theorem 8.8.2 Let L : Cc (X) → C be a positive linear functional where X is ametric space and X is a countable union of compact sets. Then there exists a completemeasure µ defined on a σ algebra F which contains the Borel sets B (X) which is finiteon compact sets and has the following properties. µ is regular. If E is measurable, thereare Fσ and Gδ sets F,G such that F ⊆ E ⊆ G and µ (G\F) = 0 so µ)F = µ (E) = µ (G).Then µ and F are uniquely determined.