DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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.

Measure-null sets and almost-everywhere statements relative to a measure

Definition

In a measure space $(X, A, μ)$ (Measure spaces), a measurable set $N \in A$ is $μ$ -null if $μ (N) = 0$ .

A property $P (x)$ holds $μ$ -almost everywhere, or for $μ$ -almost every $x$ , if its exceptional set is contained in a measurable $μ$ -null set: there is $N \in A$ with $μ (N) = 0$ such that $P (x)$ holds for every $x \in X ∖ N$ . Both notions are relative to the named measure $μ$ .

Depends on

Measure spaces

Used by

Complete measure spaces Definition
The completion domain and proposed completed set function of a measure space Definition

Dependency tree · two levels

2 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

G. Folland, Real Analysis, 2nd ed., §1.3 (standard reference, not scraped)