DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-04

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.

Density of a measurable set at a point

Let $E \subseteq R^{n}$ be Lebesgue measurable (Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ ), and let $x \in R^{n}$ . If the limit exists, the density of $E$ at $x$ is $Θ (E, x) : = \lim_{r \to 0^{+}} \frac{λ (E \cap B (x, r))}{λ (B (x, r))},$ where the denominator is positive and finite by Euclidean balls have positive finite Lebesgue measure.

When $Θ (E, x) = 1$ , we call $x$ a density-one point of $E$ ; when $Θ (E, x) = 0$ , we call $x$ a density-zero point of $E$ .

Depends on