ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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The Riemann integral functional is represented by Lebesgue measure on an interval

Example

Assume the Axiom of Countable Choice. Let $a < b$ and define $L : C ([a, b]) \to R$ by the Riemann integral $L (f) = \int_{a}^{b} f (x) d x$ . Then $L$ is positive and its RMK representing measure is Lebesgue measure restricted to $[a, b]$ .

Facts & Assumptions

Given: The Axiom of Countable Choice; continuous functions on $[a, b]$ are Riemann and Lebesgue integrable with equal integrals.

Verification

technique · direct

1.1

Linearity of the Riemann integral makes $L$ linear, and $f \geq 0$ implies $L (f) \geq 0$ . Since $[a, b]$ is compact, $C_{c} ([a, b]) = C ([a, b])$ .

given

1.2

Under the stated choice hypothesis, Lebesgue measure is regular on [given] $R$ ; its restriction to the closed subspace $[a, b]$ is finite and Radon. For every $f \in C ([a, b])$ , equality of the Riemann and Lebesgue integrals gives $L (f) = \int_{[a, b]} f d (λ ∣_{[a, b]})$ . The RMK uniqueness theorem now identifies this measure as the representing measure.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

35 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

Donald L. Cohn, Measure Theory, 2nd ed., Chapter 7 (standard reference, not scraped)