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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A locally integrable density functional is represented by g dlambda

Example

Let $n \geq 1$ , assume the Axiom of Countable Choice, and let $g \geq 0$ be locally integrable on $R^{n}$ . Then $L_{g} (f) = \int_{R^{n}} f g d λ$ is a positive linear functional on $C_{c} (R^{n})$ represented by the Radon measure $g d λ$ .

Facts & Assumptions

Given: $n \geq 1$ , the Axiom of Countable Choice, and $g \geq 0$ locally integrable.

Verification

technique · direct

1.1

If $f \in C_{c}$ has support $K$ , then $\int ∣ f g ∣ d λ \leq ∥ f ∥_{\infty} \int_{K} g d λ < \infty$ . Hence $L_{g}$ is well defined and linear; it is positive because $g \geq 0$ .

given

1.2

The density construction makes [given] $μ_{g} (E) = \int_{E} g d λ$ a Borel measure. Every compact set is contained in a ball, so local integrability makes $μ_{g}$ finite on compact sets. Moreover, $R^{n}$ is LCH, and its rational open boxes form a countable basis. The second-countable regularity theorem therefore makes $μ_{g}$ regular, hence Radon. Its defining integral gives $L_{g} (f) = \int f d μ_{g}$ , so RMK uniqueness identifies it as the representing measure.

given∎

Depends on

Used by

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Dependency tree · two levels

42 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)