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 $R^{n}$

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
The critical Riesz potential can diverge and be essentially unbounded Counterexample
A locally integrable function on ℝⁿ Definition
Measurable Beltrami coefficients and measurable conformal structures Definition
Rademacher functions on the unit interval Definition
The Beltrami coefficient and the maximal dilatation Definition
The Littlewood-Paley square function Definition
The Lusin area function for a fixed admissible kernel and aperture Definition
Dilation determines the Riesz-potential target exponent Example
Matching C¹ pieces across a hyperplane have no jump derivative Example
One-dimensional Dirichlet Green kernel on an interval Example
Piecewise-affine approximation of a measurable coefficient Example
Pullback of a measurable ellipse field under biholomorphic maps Example
Sharp Sobolev threshold for a radial power Example
A Caratheodory integrand composed with measurable functions is measurable Lemma
Near and far bounds for a Riesz potential Lemma
Sobolev functions paste across an overlap Lemma
The fundamental lemma of the calculus of variations Lemma
A Hamel coefficient has dense graph and a nonmeasurable kernel Theorem
Cut locus of a point has riemannian volume zero Theorem
Hardy–Littlewood–Sobolev fractional integration inequality Theorem

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