ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

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

Example

Counting outer measure on $R$ , equipped with its usual metric, is a metric outer measure. Hence every Borel subset is Carathéodory measurable, while in fact every subset is measurable.

Facts & Assumptions

Given: The real line and counting outer measure.

[F1]

An outer measure on a metric space is a metric outer measure when $μ^{*} (A \cup B) = μ^{*} (A) + μ^{*} (B)$ for all nonempty $A, B$ with $d (A, B) > 0$ . (Metric outer measures)

[L1]

Counting measure is an outer measure on $P (X)$ and every subset of $X$ is Carathéodory measurable. (Counting measure is an outer measure for which every subset is measurable)

[L2]

The function $d_{R} (x, y) = ∣ x - y ∣$ is a metric on $R$ , called its usual metric. (The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded)

[L3]

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

Verification

technique · direct

1.1

By [L2], $d_{R} (x, y) = ∣ x - y ∣$ is the usual metric on $R$ .

2.1

Positively separated nonempty sets are disjoint, so additivity of counting measure on disjoint sets in [L1] gives the equality required by [F1]; if either set is empty, both sides agree by the zero value. Thus counting outer measure is metric.

step 1.1F1L1algebra

3.1

Applying [L3] gives Borel measurability, while [L1] gives the stronger conclusion that every subset of $R$ is Carathéodory measurable.

step 2.1L1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

25 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 11.2 (standard reference, not scraped)