False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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False: measure-preserving transformations are invertible

Statement

The assertion that every measure-preserving probability transformation is invertible, even after restriction to an invariant conull set, is false. Assuming countable choice, doubling on the Lebesgue circle is a counterexample.

Facts & Assumptions

[F1]

Doubling preserves the completed Lebesgue probability. Doubling preserves Lebesgue measure.

[F2]

Invertibility requires a bijection; modulo-null invertibility requires one on an invariant measurable conull restriction. Invertible measure-preserving systems.

[F3]

Translation preserves Lebesgue measurability and measure. Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation.

[F4]

Finite unions of measurable null sets are null. Finite and countable subadditivity of measures.

Refutation

Given: The assertion that every measure-preserving probability transformation is invertible, even after restriction to an invariant conull set, is false. Assuming countable choice, doubling on the Lebesgue circle is a counterexample.

1.1

The map $D (x) = {2 x}$ preserves the probability in [F1], but $D (0) = D (1 / 2) = 0$ with $0 \neq 1 / 2$ . Therefore it is not injective and is not invertible in [F2]. Countable choice is the stated assumption of the Lebesgue probability supplier.

F1F2

2.1

More strongly, suppose $C$ were a measurable conull subset of $[0, 1)$ on which $D$ is injective, and put $N = [0, 1) ∖ C$ . For each $x \in [0, 1 / 2)$ the distinct points $x, x + 1 / 2$ have equal images, so at least one belongs to $N$ . Hence $[0, 1 / 2) \subseteq N \cup (N - 1 / 2)$ . By [F3] both sets on the right are measurable and null, and [F4] makes their union null. This contradicts the interval measure $1 / 2$ on the left. There is no injective conull restriction at all, in particular none satisfying the extra invariance and inverse-measurability requirements of [F2].

1.1F1F2F3F4∎

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Sources

E–W Example 2.4 (standard reference, not scraped)