TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-27

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Every monotone real function is Borel measurable

Statement

Every monotone function $f : R \to R$ in the sense of Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $R$ , with the dictionary to monotone sequences is Borel measurable.

Facts & Assumptions

Given: A monotone function $f : R \to R$ .

[L1]

A real-valued function is measurable exactly when all of its threshold sets ${x : f (x) > a}$ are measurable. (Threshold characterisations of real-valued and extended-real-valued measurability)

Proof

technique · direct

1.1given

Suppose first that $f$ is increasing. For each real $a$ , the threshold set [given] $E_{a} : = {x : f (x) > a}$ is upward closed: if $x \in E_{a}$ and $y > x$ , then $f (y) \geq f (x) > a$ , so $y \in E_{a}$ . Therefore $E_{a}$ is one of the four Borel sets $\emptyset$ , $R$ , $(c, \infty)$ , or $[c, \infty)$ for some real $c$ .

2.1step 1.1L1algebra∎

Every set named in step 1.1 is Borel, so [L1] gives that every increasing [step 1.1, L1, algebra] real function is measurable. If $f$ is decreasing, then $- f$ is increasing and hence measurable by the first half, and therefore $f$ is measurable as well.

Depends on

Used by

For a nondecreasing function, the derivative is measurable and integrable and its integral is bounded by the total increase Theorem

Dependency tree · two levels

12 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

Sheldon Axler, Measure, Integration and Real Analysis, Section 2B (standard reference, not scraped)
John K. Hunter, Measure Theory, Section 3.2 (standard reference, not scraped)