ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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Every Lebesgue measurable proper subgroup of $(R, +)$ is null, and $Z$ and $Q$ are instances

Example

Assume the Axiom of Countable Choice. If $G \leq (R, +)$ is Lebesgue measurable and proper, then $λ_{1} (G) = 0$ . In particular both $Z$ and $Q$ are Lebesgue null subgroups of the line.

Facts & Assumptions

Given: The Axiom of Countable Choice.

[L1]

A Lebesgue measurable subgroup of $(R^{n}, +)$ of positive measure is all of $R^{n}$ (A Lebesgue measurable subgroup of $(R^{n}, +)$ of positive measure is all of $R^{n}$ ).

[L2]

Every at most countable subset of $R^{n}$ is Lebesgue null (Every at most countable subset of $R^{n}$ is Lebesgue null; in particular $λ_{1} (Q) = 0$ ).

[F1]

The rationals are countably infinite ( $Q$ is countably infinite).

[F2]

The real numbers are uncountable ( $R$ is uncountable (Cantor's nested intervals, 1874)).

Verification

technique · direct

1.1

If $G \leq (R, +)$ is measurable and proper, then the contrapositive of [L1] gives $λ_{1} (G) = 0$ .

2.1

The integers and rationals are at most countable subgroups of $(R, +)$ , so [L2] gives them Lebesgue measure zero; they are proper because [F1] and [F2] show $Q \neq R$ , and $Z \subseteq Q$ .

step 1.1L2F1F2

3.1

Step 1.1 uses measurability in an essential way: it says nothing about subgroups of $R$ that are not Lebesgue measurable.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

55 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

J. Ye, L. Yu, X. Zhao, When is $A+xA=\mathbb{R}$?, Corollary 1.2 (standard reference, not scraped)
T. Tao, An Introduction to Measure Theory (GSM 126), Exercise 1.6.8 (standard reference, not scraped)

Every Lebesgue measurable proper subgroup of (R,+)(R,+) is null, and ZZ and QQ are instances