DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-11

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.

The support of a function on $R^{n}$ and its compactly supported Riemann integral

Definition

Let $n \geq 1$ . For $f : R^{n} \to R$ , its support is $supp f : = \overline{{x \in R^{n} : f (x) \neq 0}} .$ The function is compactly supported if $supp f$ is compact.

A compactly supported $f$ is compactly supported Riemann integrable if there is a nondegenerate closed rectangle $Q$ with $supp f \subseteq int Q$ such that $f ∣_{Q}$ is Riemann integrable. Its integral over Euclidean space is defined by $\int_{R^{n}} f : = \int_{Q} f .$ The value is independent of $Q$ by The Riemann integral of a compactly supported function is independent of its bounding rectangle ↗. When the support is empty, $f = 0$ and the value is $0$ .

Depends on

Dependency tree · 2 levels

Sources

A. Leibman, Multidimensional Real Analysis, §5.5

The support of a function on RnRn and its compactly supported Riemann integral

Definition

Depends on

Dependency tree · 2 levels

Sources

The support of a function on $R^{n}$ and its compactly supported Riemann integral