CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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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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An uncountable supremum of measurable indicators can be nonmeasurable

Statement refuted

That the pointwise supremum of an arbitrary family of measurable functions must remain 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 family ${1_{{t}} : t \in V}$ on that common domain.

[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)

[L2]

The indicator of a measurable set is measurable. (An indicator function is measurable exactly when its set is measurable)

Counterexample

technique · direct

1.1

Every singleton ${t}$ belongs to the trace Lebesgue sigma-algebra, so [L2] makes each $1_{{t}}$ measurable.

givenL2

2.1

Their pointwise supremum is $1_{V}$ . If $V$ belonged to the trace Lebesgue sigma-algebra, then $V = E \cap [0, 1]$ for some Lebesgue measurable $E \subseteq R$ , making $V$ Lebesgue measurable in $R$ , contrary to [L1]. Thus [L2] makes $1_{V}$ nonmeasurable, so the arbitrary-family version fails.

step 1.1L1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)