DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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.

Almost uniform convergence

Definition

Let $(X, A, μ)$ be a measure space and let $f_{n}, f : X \to R$ be measurable. The sequence $(f_{n})$ converges to $f$ almost uniformly when for every real $ε > 0$ there is a measurable set $E \in A$ with $μ (E) < ε$ such that $(f_{n})$ converges uniformly to $f$ on $X ∖ E$ .

In expanded quantifiers, this means that for every $ε > 0$ there is a measurable $E$ with $μ (E) < ε$ such that $(\forall η > 0) (\exists N \in N) (\forall n \geq N) (\forall x \in X ∖ E) ∣ f_{n} (x) - f (x) ∣ < η .$

Depends on

Used by

FALSE: Egorov's theorem holds on every measure space False statement
Almost uniform convergence implies almost-everywhere convergence and convergence in measure Theorem
Egorov's theorem Theorem

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., Section 2.4 (standard reference, not scraped)
Terence Tao, 245A Notes 4: Modes of convergence (standard reference, not scraped)