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

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.

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A Lebesgue--Stieltjes functional is represented by its Stieltjes measure

Example

Let $F : R \to R$ be increasing and right-continuous, and let $μ_{F}$ be the Lebesgue--Stieltjes measure with $μ_{F} ((a, b]) = F (b) - F (a)$ . Then $L_{F} (f) = \int f d μ_{F}$ is a positive functional on $C_{c} (R)$ represented by $μ_{F}$ .

Facts & Assumptions

Given: $μ_{F}$ is the Lebesgue--Stieltjes measure associated with $F$ .

Verification

technique · direct

1.1

The measure $μ_{F}$ is finite on compact intervals, so the displayed integral is finite for compactly supported continuous $f$ . It is linear and positive.

given

2.1

Lebesgue--Stieltjes regularity makes $μ_{F}$ Radon without changing its half-open interval convention. Thus the definition already gives a Radon representation, and RMK uniqueness says it is the representing measure.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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)