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.

A locally integrable function on $R^{n}$

Definition

Let $f : R^{n} \to C$ be a measurable function (Borel measurable and Lebesgue measurable functions on $R^{n}$ ). We say that $f$ is locally integrable on $R^{n}$ if for every Euclidean ball $B (x, r)$ with $r > 0$ (Open ball, closed ball and sphere in a metric space) one has $\int_{B (x, r)} ∣ f (y) ∣ d λ (y) < \infty,$ so the restriction of $f$ to each ball is integrable in the sense of Integrable real and complex functions, and their integrals.

The set of such functions is denoted by $L_{l o c}^{1} (R^{n})$ . When convenient, the same notation is also used for the corresponding almost-everywhere equivalence classes.

A locally integrable function on $R^{n}$

Definition

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A locally integrable function on RnRn

Definition

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Dependency tree · 2 levels

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A locally integrable function on $R^{n}$