RemarkRemark: AI-adaptedProof: 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.

Carathéodory measurability is not Carathéodory's differentiability criterion

Carathéodory measurability (Carathéodory measurable sets) is a condition on a subset $E$ of the domain of an outer measure: every test set must split additively across $E$ and its complement. The theorem Carathéodory's characterisation: $f$ is differentiable at $c$ if and only if there is $φ : A \to R$ , continuous at $c$ , with $f (x) - f (c) = φ (x) (x - c)$ for every $x \in A$ , and then $φ$ is unique and $φ (c) = f^{'} (c)$ concerns differentiability of a real function at a limit point and characterizes it by a continuous factorization. The shared name records the mathematician, not a logical relationship between the conditions.

Depends on

Used by

Nothing in the library uses this result yet.

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., Section 1.4 (standard reference, not scraped)