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Assuming countable choice, a nondecreasing right-continuous function defines a Borel measure on $R$

Statement

Assume the Axiom of Countable Choice. Let $F : R \to R$ be nondecreasing and right-continuous. Then there is a Borel measure $μ_{F}$ on $R$ , finite on compact sets in the sense of A Borel measure on $R$ that is finite on compact sets, such that

$μ_{F} ((a, b]) = F (b) - F (a) for all a < b .$

Facts & Assumptions

Given: Countable choice, a nondecreasing right-continuous function $F : R \to R$ , and its interval set function $μ_{0, F}$ on the half-open interval algebra.

[L1]

The interval set function $μ_{0, F}$ is a premeasure on the half-open interval algebra. (The Stieltjes interval set function is a premeasure)

[L2]

Assuming countable choice, a premeasure extends to a measure on the sigma-algebra it generates. (Assuming countable choice, a premeasure extends through its induced outer measure)

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is the stated assumption spent through the extension theorem [L2].

[L3]

The family of half-open intervals $(a, b]$ with $a < b$ generates the Borel sigma-algebra $B (R)$ . (Seven generating families for the Borel sigma-algebra on the real line)

[L4]

A compact subset of $R$ is bounded. (A compact subset of $R$ is closed and bounded)

Proof

technique · direct

1.1L1L2A1L3

By [L1], $μ_{0, F}$ is a premeasure, so [L2], under [A1], gives a measure $μ$ on the generated sigma-algebra $σ (H)$ extending $μ_{0, F}$ .

By [L3], that sigma-algebra is $B (R)$ , and therefore

$μ ((a, b]) = μ_{0, F} ((a, b]) = F (b) - F (a)$

for every $a < b$ . [L1, L2, L3]

2.1step 1.1L4algebra

Let $K \subseteq R$ be compact. By [L4] there is $R > 0$ with $K \subseteq [- R, R] \subseteq (- R - 1, R]$ .

So monotonicity and step 1.1 give

$μ (K) \leq μ ((- R - 1, R]) = F (R) - F (- R - 1) < + \infty .$

Thus $μ$ is finite on compact sets. [step 1.1, L4, algebra]

3.1step 1.1step 2.1

The measure $μ$ of steps 1.1 and 2.1 is therefore a Borel measure on $R$ finite on compact sets and having the prescribed half-open interval values.

This is the required $μ_{F}$ . [step 1.1, step 2.1] ∎

Depends on

Used by

Dependency tree · two levels

43 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

Gerald B. Folland, Real Analysis, 2nd ed., Theorem 1.16 (standard reference, not scraped)
John K. Hunter, Measure Theory, Theorem 2.34 (standard reference, not scraped)

Assuming countable choice, a nondecreasing right-continuous function defines a Borel measure on RR