False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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.

FALSE: every Borel measure on $R$ is finite on compact sets

Statement

False claim. Every Borel measure on $R$ is finite on compact sets. The finiteness-on-compacts hypothesis in the Lebesgue-Stieltjes correspondence is therefore a genuine hypothesis, not a consequence of being a Borel measure.

Facts & Assumptions

Given: Counting measure $#_{R}$ on $R$ .

[L1]

Counting measure is a measure on $(R, P (R))$ . (Counting measure is a measure)

[L2]

Counting measure assigns an infinite set the value $+ \infty$ . (Counting measure on an arbitrary set)

Refutation

technique · direct

1.1

By [L1], counting measure restricts to a Borel measure on $R$ . The [L1, given] compact interval $[0, 1]$ is infinite.

L1given

2.1

Therefore [L2] gives [step 1.1, L2] $#_{R} ([0, 1]) = + \infty$ . So this Borel measure is not finite on the compact set $[0, 1]$ , and the claim is false.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Gerald B. Folland, Real Analysis, 2nd ed., Section 1.5 (standard reference, not scraped)

FALSE: every Borel measure on RR is finite on compact sets