False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-09-01

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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.
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FALSE: the Borel-representative discipline in convolution is unnecessary because continuous precomposition always preserves Lebesgue measurability

Statement

False claim. The Borel-representative discipline in the convolution construction is unnecessary because Lebesgue measurability is preserved under every continuous precomposition.

Facts & Assumptions

Given: The convolution measurability seam and the published continuous- precomposition counterexample.

[L1]

The convolution page deliberately fixes Borel representatives before forming the product-space integrand (Borel representatives make the convolution integrand Borel measurable, Convolution on $L^{1} (R^{n})$ is independent of the chosen Borel representatives).

[L2]

Continuous precomposition need not preserve Lebesgue measurability (FALSE: composing a Lebesgue measurable function with a continuous map preserves measurability).

Refutation

technique · direct

1.1

Fact [L2] gives a continuous map $T$ and a Lebesgue measurable function [L2] $u$ such that $u \circ T$ is not Lebesgue measurable. So the slogan "Lebesgue measurability survives every continuous change of variables" is false.

2.1

The displayed claim relies on exactly that false slogan. Even if some [L1, step 1.1] particular maps used in convolution behave better, the blanket justification for dropping the Borel-representative discipline recorded in [L1] fails.

L1step 1.1

3.1

Therefore the displayed claim is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

Walter Rudin, Real and Complex Analysis, 3rd ed. (standard reference, not scraped)
Terence Tao, An Introduction to Measure Theory (standard reference, not scraped)