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Positive functionals on C_c(X) are integration against a Radon measure

Statement

Let $X$ be LCH and let $Λ : C_{c} (X; R) \to R$ be positive. The Radon measure $μ$ constructed above satisfies $Λ (f) = \int_{X} f d μ (f \in C_{c} (X; R)) .$

Facts & Assumptions

Given: $μ$ is the Radon measure constructed from $Λ$ .

[L1]

The compact-set formula holds. (Compact-set formula and local finiteness of the RMK measure)

Proof

technique · direct

1.1

Let $0 \leq f \in C_{c} (X)$ and choose $N$ with $f \leq N ε$ . For $1 \leq n \leq N$ set $K_{n} = {f \geq n ε}$ and $f_{n} = \min {ε, (f - (n - 1) ε)_{+}} .$ Then $f = \sum_{n = 1}^{N} f_{n}$ , $ε 1_{K_{n}} \leq f_{n} \leq ε 1_{K_{n - 1}}$ , where $K_{0} = supp f$ .

given

1.2

The upper compact estimate $0 \leq h \leq 1_{K} \Rightarrow Λ (h) \leq μ (K)$ follows directly from [L1] by comparing $h$ with functions majorizing $1_{K}$ ; the lower estimate $1_{K} \leq h \Rightarrow μ (K) \leq Λ (h)$ is [L1]. Applying these to $f_{n} / ε$ gives $ε μ (K_{n}) \leq Λ (f_{n}) \leq ε μ (K_{n - 1}) .$ The same inequalities hold for $\int f_{n} d μ$ .

2.1

Summing step 1.2 and using $μ (K_{0}) < \infty$ shows $∣ Λ (f) - \int f d μ ∣ \leq ε μ (K_{0}) .$ Letting $ε ↓ 0$ proves equality for nonnegative $f$ . Applying it to $f^{+}$ and $f^{-}$ and using linearity proves the result for every real $f \in C_{c} (X)$ .

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Donald L. Cohn, Measure Theory, 2nd ed., Chapter 7 (standard reference, not scraped)