DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

Metric outer measures

Definition

Let $(X, d)$ be a metric space and let $μ^{*}$ be an outer measure on $X$ . An outer measure on a metric space is a metric outer measure when $μ^{*} (A \cup B) = μ^{*} (A) + μ^{*} (B)$ for all nonempty $A, B$ with $d (A, B) > 0$ .

The nonempty restriction is necessary because set distance in Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space is defined only when both sets are nonempty. If either set is empty, the same equality follows separately from $μ^{*} (\emptyset) = 0$ .

Depends on

Used by

Dependency tree · two levels

19 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., Section 11.2 (standard reference, not scraped)