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.

Every monotone real function is Borel measurable

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources