PropositionStatement: 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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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.

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Every outer-null set is Carathéodory measurable

Statement

Every set of outer measure zero, and every subset of it, is Carathéodory measurable and has outer measure zero.

Facts & Assumptions

Given: An outer measure $μ^{*}$ on $X$ , a set $N \subseteq X$ with $μ^{*} (N) = 0$ , and a subset $S \subseteq N$ .

[F1]

An outer measure on a set $X$ is a function $μ^{*} : P (X) \to [0, + \infty]$ that vanishes at the empty set, is monotone, and is countably subadditive. (Outer measures)

[L1]

For every outer measure $μ^{*}$ and all $A, E \subseteq X$ , $μ^{*} (A) \leq μ^{*} (A \cap E) + μ^{*} (A ∖ E)$ . (In the Carathéodory identity, the subadditive inequality is automatic)

Proof

technique · direct

1.1

Monotonicity gives $0 \leq μ^{*} (S) \leq μ^{*} (N) = 0$ , so $μ^{*} (S) = 0$ ; for every test set $A$ , the same argument gives $μ^{*} (A \cap S) = 0$ .

F1algebra

2.1

Since $A ∖ S \subseteq A$ , monotonicity gives $μ^{*} (A) \geq μ^{*} (A ∖ S) = μ^{*} (A \cap S) + μ^{*} (A ∖ S)$ by step 1.1; [L1] supplies the reverse inequality, so $S$ is Carathéodory measurable.

step 1.1L1algebra∎

Depends on

Used by

Carathéodory's theorem: measurable sets form a sigma-algebra carrying a complete measure Theorem

Dependency tree · two levels

5 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

T. Tao, An Introduction to Measure Theory, Exercise 1.7.1 (standard reference, not scraped)
G. Folland, Real Analysis, 2nd ed., Theorem 1.11 (standard reference, not scraped)