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Lebesgue measure is the Lebesgue-Stieltjes measure of the identity function

Statement

Assume the Axiom of Countable Choice. Let ${id}_{R} (x) : = x$ . The Lebesgue-Stieltjes measure attached to ${id}_{R}$ agrees with Lebesgue measure $λ$ from Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ on every Borel subset of $R$ .

Facts & Assumptions

Given: Countable choice and the identity function ${id}_{R} : R \to R$ .

[L1]

Assuming countable choice, every nondecreasing right-continuous function on $R$ defines a Borel measure through the Lebesgue-Stieltjes construction. (Assuming countable choice, a nondecreasing right-continuous function defines a Borel measure on $R$ )

[L2]

For every half-open interval $(a, b] \subseteq R$ , Lebesgue measure satisfies $λ ((a, b]) = b - a$ . (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included)

[L3]

A Borel measure on $R$ finite on compact sets is uniquely determined by its values on half-open intervals. (The interval data on $(a, b]$ determines the Borel measure uniquely)

Proof

technique · direct

1.1L1

The identity function is nondecreasing and right-continuous, so [L1] gives a Borel measure $μ_{id}$ with

$μ_{id} ((a, b]) = {id}_{R} (b) - {id}_{R} (a) = b - a .$

[L1]

1.2L2

By [L2], Lebesgue measure has the same half-open interval values: $λ ((a, b]) = b - a$ for every $a < b$ .

2.1step 1.1step 1.2L3

The measures $μ_{id}$ and $λ$ are both Borel and finite on compact sets, and by steps 1.1 and 1.2 they agree on every half-open interval.

Therefore [L3] gives $μ_{id} = λ$ . [step 1.1, step 1.2, L3] ∎

Depends on

Used by

The identity function generates Lebesgue measure Example
FALSE: every measure is absolutely continuous or singular with respect to Lebesgue measure False statement
FALSE: the Radon-Nikodym theorem holds without sigma-finiteness False statement

Dependency tree · two levels

25 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

John K. Hunter, Measure Theory, Example 2.35 (standard reference, not scraped)