CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck 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.

A function can have measurable level sets without being measurable

Statement refuted

That measurability of every level set ${x : f (x) = a}$ is enough to make $f$ measurable.

Facts & Assumptions

Given: The Axiom of Choice, the Lebesgue measurable space $([0, 1], L (R) ∣_{[0, 1]})$ , a Vitali set $V \subseteq [0, 1]$ , and the function $f : [0, 1] \to R$ defined by $f (x) = x$ for $x \notin V$ , $f (x) = x + 2$ for $x \in V$ .

[L1]

Assuming the Axiom of Choice, Vitali sets exist and are not Lebesgue measurable. (Assuming choice on the cosets of $Q$ in $R$ , a Vitali set in $[0, 1]$ exists, Assuming the Axiom of Choice, a Vitali set is not Lebesgue measurable)

Counterexample

technique · direct

1.1given

Every level set of $f$ is empty, a singleton, or a two-point set, so every [given] level set is measurable.

2.1step 1.1L1∎

Yet $f^{- 1} ([2, 3]) = V$ , and [L1] says that $V$ is not measurable. Therefore [step 1.1, L1] $f$ is not measurable.

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)