False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

Every translation-invariant measure on the Borel sets of $R$ is a nonnegative multiple of Lebesgue measure

Statement

Assume the Axiom of Countable Choice. Every translation-invariant measure on the Borel sets of $R$ is a nonnegative multiple of Lebesgue measure.

Facts & Assumptions

Given: The Axiom of Countable Choice and counting measure $#$ on $R$ , restricted to the Borel sigma-algebra.

[L1]

Assuming countable choice, a translation-invariant measure on the Borel sets of $R^{n}$ giving the unit cube measure one is the restriction of Lebesgue measure (A translation-invariant measure on the Borel sets of $R^{n}$ giving the unit cube measure one is the restriction of Lebesgue measure).

[F1]

The counting set function on $P (X)$ is the map sending a finite subset to its cardinality and an infinite subset to $+ \infty$ (Counting measure on an arbitrary set).

[F2]

For every set $X$ , the counting set function $#_{X}$ is a measure on $(X, P (X))$ (Counting measure is a measure).

[L2]

Assuming countable choice, a box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

Refutation

technique · direct

1.1

By [F1] and [F2], counting measure restricted to the Borel sets of $R$ is a measure. It is translation invariant because for every real $a$ the map $x \mapsto x + a$ is a bijection of $R$ , so a Borel set and its translate have the same finite cardinality or are both infinite.

F1F2

2.1

It gives the singleton ${0}$ the value $1$ and the unit interval $[0, 1]$ the value $+ \infty$ , while [L2] gives Lebesgue measure $0$ for ${0}$ and $1$ for $[0, 1]$ ; so no nonnegative scalar multiple of Lebesgue measure equals counting measure.

step 1.1L2algebra

3.1

This does not contradict [L1], because [L1] fixes the value of the unit cube and thereby forces finiteness on bounded sets, which the false statement omits.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

38 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

T. Tao, An Introduction to Measure Theory (GSM 126), Exercise 1.2.23 (standard reference, not scraped)

Every translation-invariant measure on the Borel sets of RR is a nonnegative multiple of Lebesgue measure