188 CHAPTER 8. POSITIVE LINEAR FUNCTIONALS

Proof: Let K1 = K \∪ni=2Vi. Thus K1 is compact and K1 ⊆ V1. Let K1 ⊆W1 ⊆W 1 ⊆

V1 with W 1compact. To obtain W1, use Theorem 8.1.3 to get f such that K1≺ f ≺V1 and letW1≡{x : f (x) ̸= 0} .Thus W1,V2, · · ·Vn covers K and W 1⊆V1. Let K2 =K \(∪n

i=3Vi∪W1).Then K2 is compact and K2 ⊆ V2. Let K2 ⊆W2 ⊆W 2 ⊆ V2 W 2 compact. Continue thisway finally obtaining W1, · · · ,Wn, K ⊆W1 ∪ ·· · ∪Wn, and W i ⊆ Vi W i compact. Now letW i ⊆Ui ⊆U i ⊆Vi ,U i compact.

Wi Ui Vi

By Theorem 8.1.3, let U i ≺ φ i ≺Vi, ∪ni=1W i ≺ γ ≺ ∪n

i=1Ui. Define

ψ i(x) ={

γ(x)φ i(x)/∑nj=1 φ j(x) if ∑

nj=1 φ j(x) ̸= 0,

0 if ∑nj=1 φ j(x) = 0.

If x is such that ∑nj=1 φ j(x) = 0, then x /∈ ∪n

i=1U i. Consequently γ(y) = 0 for all y near xand so ψ i(y) = 0 for all y near x. Hence ψ i is continuous at such x. If ∑

nj=1 φ j(x) ̸= 0, this

situation persists near x and so ψ i is continuous at such points from the top description ofψ i. Therefore ψ i is continuous. If x ∈ K, then γ(x) = 1 and so ∑

nj=1 ψ j(x) = 1. Clearly

0≤ψ i (x)≤ 1 and spt(ψ j)⊆Vj. As to the last claim, keep Vi the same but replace Vj, j ̸= iwith Ṽj ≡Vj \H. Now in the proof above, applied to this modified collection of open sets,if j ̸= i,φ j (x) = 0 whenever x ∈ H. Therefore, ψ i (x) = 1 on H. ■

8.2 Positive Linear Functionals and MeasuresNow with this preparation, here is the main result called the Riesz representation theoremfor positive linear functionals. I am presenting this for a metric space, but in this book, wewill typically have X = Rp.

Theorem 8.2.1 (Riesz representation theorem) Let L be a positive linear functionalon Cc(X) where (X ,d) is a metric space having closed balls compact. Thus L f ∈ C if f ∈Cc (X). Then there exists a σ algebra F containing the Borel sets and a unique measureµ , defined on F, such that

µ is complete, (8.2)µ(K) < ∞ for all K compact, (8.3)

µ(F) = sup{µ(K) : K ⊆ F, K compact}, (8.4)

for all F ∈F ,µ(F) = inf{µ(V ) : V ⊇ F, V open} (8.5)

for all F ∈F, and ∫f dµ = L f for all f ∈Cc(X). (8.6)

This extends the functional L because the integral will be defined for all f ∈ L1 (X) ingeneral, a much larger set than Cc (X). The two assertions 8.4 and 8.5 are called respectively