PropositionStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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

Statement

Let $n \geq 1$ and assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Every at most countable subset $E \subseteq R^{n}$ (Finite, countably infinite, countable, uncountable) is Lebesgue measurable with

$λ_{n} (E) = 0,$

so $E$ is a $λ_{n}$ -null set (Measure-null sets and almost-everywhere statements relative to a measure). In particular every singleton is null, and on the real line the set $Q_{R}$ of rational reals (The rationals embed densely in the reals) satisfies $λ_{1} (Q_{R}) = 0$ .

Facts & Assumptions

Given: A natural number $n \geq 1$ , the Axiom of Countable Choice, and an at most countable set $E \subseteq R^{n}$ .

[L1]

Every set $R$ with $R^{\circ} \subseteq R \subseteq \overline{R}$ is Lebesgue measurable with $λ_{n} (R) = \prod_{i < n} (b_{i} - a_{i})$ , and this gives measure $0$ to all of them whenever $a_{i} = b_{i}$ for some $i < n$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[L2]

Assuming countable choice, $L (R^{n})$ is a sigma-algebra and $λ_{n}$ is a complete measure on it (Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume).

[F1]

$A$ is at most countable if it is finite or countably infinite (Finite, countably infinite, countable, uncountable); a nonempty $A$ is at most countable if and only if there is a surjection $s : N \to A$ (A nonempty set is at most countable iff it is a surjective image of $N$ ).

[F2]

For a measure $μ$ and measurable $(E_{k})_{k \in N}$ , $μ (⋃_{k \in N} E_{k}) \leq \sum_{k = 0}^{\infty} μ (E_{k})$ (Finite and countable subadditivity of measures).

[F3]

$Q \approx N$ : the rationals are countably infinite ( $Q$ is countably infinite), and $Q_{R}$ denotes the image of $Q$ in $R$ under the canonical order-preserving field embedding (The rationals embed densely in the reals).

[F4]

A measurable set $N \in A$ is $μ$ -null if $μ (N) = 0$ (Measure-null sets and almost-everywhere statements relative to a measure); a sigma-algebra is closed under countable unions (Sigma-algebras).

[F5]

$[a, b] : = {x \in R^{m} : a_{j} \leq x_{j} \leq b_{j} (j < m)}$ (Axis-parallel rectangles in $R^{m}$ and their volume).

Proof

technique · direct

1.1L1F5

A singleton ${x} \subseteq R^{n}$ is the closed rectangle $[x, x]$ , whose sides all satisfy $a_{i} = b_{i} = x_{i}$ , so it is Lebesgue measurable with $λ_{n} ({x}) = 0$ .

1.2L2F4

The empty set is Lebesgue measurable with measure $0$ .

2.1step 1.1L2F1F2F4

Let $E$ be nonempty and at most countable and fix a surjection $s : N \to E$ ; then $E = ⋃_{k \in N} {s (k)}$ is a countable union of measurable sets, hence measurable, and countable subadditivity gives $λ_{n} (E) \leq \sum_{k = 0}^{\infty} λ_{n} ({s (k)}) = 0$ .

3.1step 1.2step 2.1F3∎

Steps 1.2 and 2.1 cover both cases, and $Q_{R}$ is a countably infinite subset of $R$ , so $λ_{1} (Q_{R}) = 0$ .

Depends on

Used by

Polar integration may discard the cut locus Corollary
The graph of a measurable function Rⁿ to R is Lebesgue null Corollary
Lp and Sobolev classes do not determine point values Counterexample
Mod-null invariance need not be strict invariance Counterexample
Pointwise modification can destroy path continuity Counterexample
ℚ∩[0,1] is Lebesgue null and has Jordan outer content one Counterexample
The complementarity product needs extra regularity Counterexample
The Dirichlet function is positive on a dense set but has Lebesgue integral 0 Counterexample
The fundamental Hessian is not absolutely locally integrable Counterexample
Weakly measurable need not be strongly measurable Counterexample
Add, cov, non and cof for null and meagre ideals Definition
Complex Lp classes and Euclidean test-function conventions Definition
Mihlin smoothness convention above half the dimension Definition
Newtonian potential of compactly supported data Definition
A boundary atom gives an h1 function without an L1 density Example
A dense G_δ subset of ℝ of Lebesgue measure zero containing every rational, and its meager complement of full measure Example
Every Lebesgue measurable proper subgroup of (ℝ,+) is null, and ℤ and ℚ are instances Example
Kac mean return to a half-circle under irrational rotation Example
Normal approximation to binomial probabilities Example
Quantile coupling on the real line Example
The Gauss map preserves Gauss measure Example
The L¹_loc class of 1_ℚ has every point as a Lebesgue point Example
The Lebesgue measure of an interval, of a box, of ℚ and of the irrationals in [0,1] Example
The one-dimensional obstacle reaction is supported on the contact set Example
The regular representation of the real line as a multiplicity-one integral of characters Example
FALSE: a dense subset of ℝ of outer measure zero and a dense subset of full inner measure cannot both meet every open interval False statement
False: an ergodic invariant sigma-algebra has only two sets False statement
Lebesgue outer measure agrees with Jordan outer content on every bounded subset of ℝⁿ False statement
Base-b digit cylinders are orbit cylinders Lemma
Interval realization from refining small diameter partitions Lemma
Kolmogorov atomic kernel maxima Lemma
Local integrability of the Laplace fundamental kernel Lemma
The unit sphere is Lebesgue null Lemma
Transfer of null and meagre invariants between Cantor space and the line Lemma
The published refutations separating nullity from nowhere density hold verbatim for Lebesgue measure Remark
Gibbs overshoot at a piecewise C¹ jump Theorem
Green representation for classical Poisson data Theorem
Newtonian potentials solve the distributional Poisson equation Theorem

Dependency tree · two levels

67 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 (UC Davis lecture notes), Example 2.3 (standard reference, not scraped)
T. Tao, An Introduction to Measure Theory (GSM 126), Section 1.2 (standard reference, not scraped)

Every at most countable subset of RnRn is Lebesgue null; in particular λ1(Q)=0λ1​(Q)=0