DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Borel measurable and Lebesgue measurable functions on $R^{n}$

Definition

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), so Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume makes $L (R^{n})$ a sigma-algebra. Let $n, m \in N$ with $n, m \geq 1$ .

A function $f : R^{n} \to R^{m}$ is Borel measurable when it is measurable as a map

$f : (R^{n}, B (R^{n})) \to (R^{m}, B (R^{m}))$

in the sense of A measurable function between measurable spaces.
A function $f : R^{n} \to R^{m}$ is Lebesgue measurable when it is measurable as a map

$f : (R^{n}, L (R^{n})) \to (R^{m}, B (R^{m})),$

where $L (R^{n})$ is the Lebesgue sigma-algebra of Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ .

When $m = 1$ , these are the corresponding notions for real-valued functions.

Depends on

Used by

Continuous functions on Euclidean spaces are Borel measurable Corollary

Dependency tree · two levels

27 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, Definition 3.3 (standard reference, not scraped)

Borel measurable and Lebesgue measurable functions on RnRn