ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge 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.

The zero-one outer measure on a two-point set has only the trivial measurable sets

Example

On $X = {0, 1}$ , define $μ^{*} (\emptyset) = 0$ and $μ^{*} (E) = 1$ for every nonempty $E \subseteq X$ . This is an outer measure, and its Carathéodory measurable sets are exactly $\emptyset$ and $X$ .

Facts & Assumptions

Given: The finite set and set function in the Example.

[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)

[F2]

A set $E \subseteq X$ is Carathéodory measurable for $μ^{*}$ when $μ^{*} (A) = μ^{*} (A \cap E) + μ^{*} (A ∖ E)$ for every $A \subseteq X$ . (Carathéodory measurable sets)

Verification

technique · direct

1.1F1algebra

Exhausting the four subsets gives normalization and monotonicity. For countable subadditivity, an empty union has value $0$ , while a nonempty union has value $1$ and contains a point lying in a nonempty cover member, so the covering sum is at least $1$ ; hence [F1] holds.

2.1step 1.1F2algebra∎

The sets $\emptyset$ and $X$ satisfy [F2] identically. For either singleton $E$ , the test $A = X$ gives $1 = μ^{*} (X)$ but $μ^{*} (A \cap E) + μ^{*} (A ∖ E) = 1 + 1$ , so neither singleton is measurable.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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