DefinitionDefinition: AI-adaptedProof: 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.

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

Definition

Let $(X, A, μ)$ be a measure space. Its completion domain is

$\overline{A} : = {E \subseteq X : E = A \cup N for some A, Z \in A and N \subseteq Z with μ (Z) = 0} .$

For a displayed representation $E = A \cup N$ , define the proposed completed set function by

$\overline{μ} (E) : = μ (A) .$

The value must not depend on the representation. That obligation is discharged by The completed measure is independent of the representing measurable set ↗, and Assuming countable choice, the completion domain is a sigma-algebra ↗ proves, under the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), that $\overline{A}$ is a sigma-algebra. Thus the definite phrases completion sigma-algebra and completed measure are used only with those two results in force. Under the same choice hypothesis, Assuming countable choice, every measure space has a unique complete extension to its completion ↗ proves that $\overline{μ}$ is the unique complete extension on this domain.

Depends on

Used by

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

G. Folland, Real Analysis, 2nd ed., Theorem 1.9 (standard reference, not scraped)