TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Every Borel set is Carathéodory measurable for a metric outer measure

Statement

Every Borel subset of a metric space is Carathéodory measurable for every metric outer measure.

Facts & Assumptions

Given: A metric space $(X, d)$ , a metric outer measure $μ^{*}$ , and its Carathéodory family $M_{μ^{*}}$ .

[L1]

Every closed subset of a metric space is Carathéodory measurable for every metric outer measure. (Closed sets are Carathéodory measurable for metric outer measures)

[L2]

The Carathéodory measurable subsets form a sigma-algebra, and the restriction of the outer measure to it is a complete measure. (Carathéodory's theorem: measurable sets form a sigma-algebra carrying a complete measure)

[F1]

The Borel sigma-algebra $B (X)$ of a topological space is the sigma-algebra generated by its open sets. (The Borel sigma-algebra of a topological space)

[L3]

For every $E \subseteq P (X)$ , $σ_{X} (E)$ is the unique smallest sigma-algebra on $X$ containing $E$ . (Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal)

Proof

technique · direct

1.1

By [L2], $M_{μ^{*}}$ is a sigma-algebra. It contains every closed set by [L1], and hence contains every open set by closure under complements.

L1L2algebra

2.1

By [F1], the Borel sigma-algebra is generated by the open sets, so minimality in [L3] and step 1.1 give $B (X) \subseteq M_{μ^{*}}$ ; this also covers the empty metric space and the one-point space.

step 1.1F1L3∎

Depends on

Used by

Counting outer measure is a metric outer measure on the real line Example

Dependency tree · two levels

20 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., Proposition 11.16 (standard reference, not scraped)