516 CHAPTER 19. HAUSDORFF SPACES AND MEASURES

From the picture as needed: Let µ (U \F) < ε where U is open and let K ⊆ U andµ (U \K)< ε , µ(V \ (U \F))< ε with V open and V ⊇U \F =U ∩FC so VC ⊆UC ∪F .This is possible because all sets are in S . Then VC∩K ⊆

(UC ∪F

)∩K = F∩K ⊆ F. Now

VC ∩K is compact and

µ(F \(K∩VC)) = µ

(F ∩

(KC ∪V

))= µ (F ∩V )+µ

(F ∩KC)

≤ µ (F ∩V )+µ (U \K)< µ (F ∩V )+ ε (19.6)

However,

ε > µ(V \ (U \F)) = µ

(V ∩

(U ∩FC)C)= µ

(V ∩

(UC ∪F

))≥ µ (V ∩F)

and so from 19.6, µ(F \(K∩VC

))≤ 2ε . Since K∩VC is compact, this shows 19.5. ■

19.7 Measures and Positive Linear FunctionalsThis is on the Riesz representation theorem for positive linear functionals. It is a reallymarvelous result. It produces measures on locally compact Hausdorff spaces. Thus thisdoesn’t help a lot in producing measures on infinite dimensional spaces but it works greaton Rn or closed subsets of Rn and so forth.

Definition 19.7.1 Let (Ω,τ) be a topological space. L : Cc(Ω)→ C is called apositive linear functional if L is linear, L(a f1+b f2) = aL f1+bL f2, and if L f ≥ 0 wheneverf ≥ 0.

Theorem 19.7.2 (Riesz representation theorem) Let (Ω,τ) be a locally compactHausdorff space and let L be a positive linear functional on Cc(Ω). Then there exists a σ

algebra S containing the Borel sets and a unique measure µ , defined on S , such that

µ is complete, (19.7)µ(K) < ∞ for all K compact, (19.8)

µ(F) = sup{µ(K) : K ⊆ F, K compact},

for all F open and for all F ∈S with µ(F)< ∞,

µ(F) = inf{µ(V ) : V ⊇ F, V open}

for all F ∈S , and ∫f dµ = L f for all f ∈Cc(Ω). (19.9)

The plan is to define an outer measure and then to show that it, together with the σ

algebra of sets measurable in the sense of Caratheodory, satisfies the conclusions of thetheorem. Always, K will be a compact set and V will be an open set.

Definition 19.7.3 µ(V )≡ sup{L f : f ≺V} for V open,

µ( /0) = 0, µ(E)≡ inf{µ(V ) : V ⊇ E}

for arbitrary sets E.