ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-30

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 signed measure with density sin x on [0,2pi] exhibits the nonuniqueness of Hahn decompositions

Example

Let $λ$ be Lebesgue measure on $[0, 2 π]$ and define $ν (E) : = \int_{E} \sin x d λ (E \in B ([0, 2 π])) .$ Then $P_{0} = (0, π), N_{0} = {0} \cup [π, 2 π]$ is a Hahn decomposition, and so is $P_{1} = [0, π], N_{1} = (π, 2 π],$ because the points $0$ and $π$ are $ν$ -null.

Facts & Assumptions

Given: The signed measure $ν (E) = \int_{E} \sin x d λ$ on $[0, 2 π]$ .

[L1]

A real $L^{1}$ density defines a finite signed measure whose canonical Hahn sets are ${f > 0}$ and ${f \leq 0}$ . (A real L^1 density defines a finite signed measure with its canonical Hahn and Jordan data)

[L2]

Hahn decompositions are unique only up to null sets. (Hahn decomposition for signed measures, unique up to total-variation-null sets)

[A1]

On $[0, 2 π]$ , one has $\sin x > 0$ on $(0, π)$ , $\sin x < 0$ on $(π, 2 π)$ , and $\sin x = 0$ exactly at $0, π, 2 π$ .

Verification

technique · direct

1.1

By [A1], the density theorem [L1] makes $(0, π)$ positive and ${0} \cup [π, 2 π]$ negative for $ν$ . Thus $P_{0}, N_{0}$ is a Hahn decomposition.

L1A1

2.1

The points $0$ and $π$ are $ν$ -null because $\sin x = 0$ there. Moving those null points from $N_{0}$ to $P_{1}$ preserves positivity and negativity, so $P_{1}, N_{1}$ is another Hahn decomposition. It is distinct from $P_{0}, N_{0}$ and compatible with the uniqueness clause of [L2].

L2A1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Richard F. Bass, Real Analysis for Graduate Students, Example 12.3 (standard reference, not scraped)