PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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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.
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Doubling preserves Lebesgue measure

Statement

Assume countable choice. The doubling map $D (x) = {2 x}$ is a continuous, surjective, non-injective transformation preserving Borel Lebesgue probability on the circle and its completion.

Facts & Assumptions

[F1]

The preservation theorem applies to every integer b>=2. Integer-base circle maps preserve Lebesgue measure.

[F2]

The stable doubling map has the fractional-part formula. The circle, rotations and the doubling map.

Proof

Given: Assume countable choice. The doubling map $D (x) = {2 x}$ is a continuous, surjective, non-injective transformation preserving Borel Lebesgue probability on the circle and its completion.

1.1

By the definitions, $D = D_{2}$ . Since 2 is an allowed integer base, the base-map theorem proves preservation on both sigma-algebras and continuity. Its countable-choice measure and completion hypothesis is the assumption here.

F1F2

2.1

Explicitly $D^{- 1} [a, c) = [a / 2, c / 2) \cup [(a + 1) / 2, (c + 1) / 2)$ for $0 \leq a < c \leq 1$ ; the two pieces have total length c-a. For each y, y/2 is a preimage, while $D (0) = D (1 / 2) = 0$ exhibits failure of injectivity. These branch identities also show why preservation concerns inverse images.

step 1.1F2∎

Depends on

Used by

Mod-null invariance need not be strict invariance Counterexample
Dyadic preimages under doubling Example
Recurrence to a dyadic interval under doubling Example
False: measure-preserving transformations are invertible False statement
Doubling is ergodic for Lebesgue measure Theorem

Dependency tree · two levels

18 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

E–W Example 2.4 pp.14–15 (standard reference, not scraped)