DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Luzin's property $(N)$ on a compact interval

Definition

For $a \leq b$ and $F : [a, b] \to R$ , say that $F$ has Luzin's property $(N)$ if, for every $E \subseteq [a, b]$ , $λ^{*} (E) = 0$ implies $λ^{*} (F (E)) = 0$ , where $λ^{*}$ is Lebesgue outer measure (Lebesgue outer measure on $R^{n}$ ). This formulation is meaningful before any measurability of $F (E)$ has been proved. It is an image condition; it asserts nothing about preimages of null sets. On $[a, a]$ it holds automatically.

Depends on

Lebesgue outer measure on $\mathbb{R}^n$

Used by

Luzin's property (N) does not imply absolute continuity Counterexample
The Cantor function fails Luzin's property (N) Counterexample
Luzin's property (N) implies absolute continuity False statement
Luzin's property (N) gives an integral growth estimate Lemma
Absolutely continuous functions have Luzin's property (N) Theorem
Banach--Zarecki characterisation of absolute continuity Theorem

Dependency tree · two levels

5 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

Christopher Heil, Absolute Continuity and the Banach--Zaretsky Theorem, §3.4 (standard reference, not scraped)

Luzin's property (N)(N) on a compact interval