LemmaStatement: AI-adaptedProof: AI-adaptedprecheck 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.

In the Carathéodory identity, the subadditive inequality is automatic

Statement

For every outer measure $μ^{*}$ and all $A, E \subseteq X$ , $μ^{*} (A) \leq μ^{*} (A \cap E) + μ^{*} (A ∖ E)$ . Consequently, to prove that $E$ is Carathéodory measurable (Carathéodory measurable sets), it suffices to prove the reverse inequality for every $A \subseteq X$ .

Facts & Assumptions

Given: An outer measure $μ^{*}$ on $X$ and subsets $A, E \subseteq X$ .

[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)

Proof

technique · direct

1.1givenalgebra

The test set decomposes as $A = (A \cap E) \cup (A ∖ E)$ .

2.1step 1.1F1algebra∎

Apply countable subadditivity in [F1] to the two sets in step 1.1 followed by empty sets. This gives $μ^{*} (A) \leq μ^{*} (A \cap E) + μ^{*} (A ∖ E)$ , including $0 = 0 + 0$ when $A = \emptyset$ and without subtracting when a value is $+ \infty$ .

Depends on

Dependency tree · 2 levels

Sources

G. Folland, Real Analysis, 2nd ed., Section 1.4