ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

Two different normalizations give the same Lebesgue-Stieltjes measure

Example

If $G (x) = F (x) + 3$ , then $F$ and $G$ define the same Lebesgue-Stieltjes measure. This is the additive-constant ambiguity that the normalization $F_{μ} (0) = 0$ removes.

Facts & Assumptions

Given: Countable choice, a nondecreasing right-continuous function $F : R \to R$ , and the shifted function $G : = F + 3$ .

[L1]

Assuming countable choice, two nondecreasing right-continuous functions define the same Lebesgue-Stieltjes measure exactly when their difference is constant. (Assuming countable choice, finite-on-compacts Borel measures on $R$ correspond to nondecreasing right-continuous functions modulo constants)

Verification

technique · direct

1.1

The function $G$ is nondecreasing and right-continuous whenever $F$ is, and [given] $G - F$ is the constant function $3$ .

given

2.1

Therefore [L1] gives $μ_{G} = μ_{F}$ . The two distribution functions are [step 1.1, L1] distinct unless $3 = 0$ , so the normalization convention is doing real work.

step 1.1L1∎

Depends on

Assuming countable choice, finite-on-compacts Borel measures on $\mathbb{R}$ correspond to nondecreasing right-continuous functions modulo constants

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Gerald B. Folland, Real Analysis, 2nd ed., Theorem 1.16 (standard reference, not scraped)