DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-01

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 Riemann integral of a bounded function over a bounded Jordan measurable set

Definition

Let $E \subseteq R^{m}$ be bounded in the metric sense of Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space and Jordan measurable, and let $f : E \to R$ be bounded. Choose a nondegenerate rectangle $Q \supseteq E$ , whose existence follows from Jordan inner and outer content and Jordan measurable bounded sets in $R^{m}$ , and define the zero extension ${\tilde{f}}_{Q} (x) : = {\begin{cases} f (x), & x \in E, \\ 0, & x \in Q ∖ E . \end{cases}$ The function $f$ is Riemann integrable over $E$ when ${\tilde{f}}_{Q}$ is integrable over $Q$ , and then $\int_{E} f : = \int_{Q} {\tilde{f}}_{Q} .$ Independence of the bounding rectangle, for both integrability and value, is proved in The Riemann integral over a Jordan set is independent of the bounding rectangle ↗ and recorded as the definition's forward justification. For $f = 1$ , the zero extension is $1_{E}$ , so A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content gives $\int_{E} 1 = cont (E)$ .

Depends on

Dependency tree · 2 levels

Sources