DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 $R^{n}$

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