CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-31

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 Radon-Nikodym derivatives can differ on a null set

Statement refuted

Assume the Axiom of Countable Choice. The Radon-Nikodym derivative is a uniquely determined function.

Facts & Assumptions

Given: Countable choice, the zero measure on $(R, B (R))$ , and the Cantor set $C$ .

[L1]

The Cantor set is Lebesgue measurable and Lebesgue null. (The Cantor set is an uncountable subset of $R$ of Lebesgue measure zero)

[L2]

The integral of a nonnegative function over a null set vanishes. (A nonnegative integral over a null set vanishes)

[L3]

A Radon-Nikodym derivative is only an almost-everywhere equivalence class of representing functions. (The Radon-Nikodym derivative as an almost-everywhere equivalence class)

Counterexample

technique · direct

1.1

Let $h_{0} : = 0$ and $h_{1} : = χ_{C}$ . For every measurable set $E$ , [L1] and [L2] give $\int_{E} h_{1} d λ = λ (E \cap C) = 0 = \int_{E} h_{0} d λ .$ Thus both $h_{0}$ and $h_{1}$ represent the zero measure relative to $λ$ .

L1L2construct

2.1

The functions $h_{0}$ and $h_{1}$ are not equal pointwise because $h_{1} = 1$ on $C$ , but [L3] says only almost-everywhere equality is required. Hence pointwise uniqueness fails.

step 1.1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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.