False statementConstruction: Literature-sourcedVerification: AI-adaptedverified 2026-09-26 (gpt-6-sol)

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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.

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FALSE: if the absolute value is measurable, then the function is measurable

Statement

False claim. If $∣ f ∣$ is measurable, then $f$ is measurable.

Facts & Assumptions

Given: The Axiom of Choice, the Lebesgue measurable-space structure on $[0, 1]$ , and a Vitali set $V \subseteq [0, 1]$ .

[L1]

Assuming the Axiom of Choice, Vitali sets exist. (Assuming choice on the cosets of $Q$ in $R$ , a Vitali set in $[0, 1]$ exists)

[L2]

Assuming the Axiom of Choice, a Vitali set is not Lebesgue measurable. (Assuming the Axiom of Choice, a Vitali set is not Lebesgue measurable)

Refutation

technique · direct

1.1givenL1

By [L1], choose a Vitali set $V \subseteq [0, 1]$ . Define $f : [0, 1] \to R$ by $f (x) = 1$ for $x \in V$ and $f (x) = - 1$ for $x \in [0, 1] ∖ V$ . Then $∣ f ∣$ is the constant function $1$ , hence measurable.

2.1step 1.1L2∎

The inverse image $f^{- 1} ((0, \infty)) = {x : f (x) > 0} = V$ is not Lebesgue measurable by [L2]. Since $(0, \infty)$ is open, $f$ is not measurable even though $∣ f ∣$ is.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Sheldon Axler, Measure, Integration and Real Analysis, Exercise 29 (standard reference, not scraped)