False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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FALSE: every section of a completed-product measurable function is measurable

Statement

If $f$ is measurable for the completed product $\overline{μ \times ν}$ , then $f_{x}$ is measurable for every $x$ and $f^{y}$ is measurable for every $y$ .

Facts & Assumptions

Given: Lebesgue measure $λ$ on $R$ , a non-Lebesgue-measurable set $N \subseteq R$ , the set $E : = {0} \times N \subseteq R^{2}$ , and the indicator function $f : = 1_{E}$ .

[L1]

For completed products, section measurability is guaranteed only for almost every parameter. (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability)

[L2]

The completed product sigma-algebra contains every subset of a $(λ \times λ)$ -null set. (The completed product measure)

[L3]

On measurable rectangles, the product measure satisfies $(λ \times λ) (A \times B) = λ (A) λ (B)$ . (For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique)

Refutation

technique · direct

1.1

For each $n \geq 1$ , the rectangle $Z_{n} : = {0} \times [- n, n]$ satisfies $(λ \times λ) (Z_{n}) = λ ({0}) λ ([- n, n]) = 0$ by [L3], so $Z : = {0} \times R = ⋃_{n \geq 1} Z_{n}$ is $(λ \times λ)$ -null.

L3algebra

2.1

Because $E \subseteq Z$ , [L2] puts $E$ in the completed product sigma-algebra, so $f = 1_{E}$ is measurable for the completed product.

L2step 1.1

3.1

The section at $0$ is $f_{0} = 1_{N}$ , which is not measurable because $N$ is not Lebesgue measurable. Thus the displayed claim fails even though $f$ is measurable for the completed product, and [L1] is sharp.

L1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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

John K. Hunter, Measure Theory, Example 5.20 (standard reference, not scraped)