False statementConstruction: Literature-sourcedVerification: 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: if every level set of a real-valued function is measurable, then the function is measurable

Statement

False claim. If every level set ${x : f (x) = a}$ of a real-valued function is measurable, then the function itself 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 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

1.1

Define $f : [0, 1] \to R$ by [given, L1] $f (x) = x$ for $x \notin V$ and $f (x) = x + 2$ for $x \in V$ . Every level set of $f$ is empty, a singleton, or a two-point set, hence measurable.

givenL1

2.1

The set $f^{- 1} ([2, 3])$ is exactly $V$ , because on $[0, 1] ∖ V$ the [step 1.1, L2] values lie in $[0, 1]$ and on $V$ they lie in $[2, 3]$ . By [L2], the set $V$ is not measurable, so $f$ is not measurable.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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)