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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)

[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

1.1

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

L1L2L3

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]

2.1

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

step 1.1L4algebra

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]

3.1

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.

step 1.1step 2.1

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

Depends on

Used by

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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