TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

How statement and proof provenance work

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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation

Statement

Let $n \geq 1$ , let $h \in R^{n}$ , and let $E + h$ be the translate of $E \subseteq R^{n}$ (Translation of a subset of $R^{n}$ ). Then:

$λ_{n}^{*} (E + h) = λ_{n}^{*} (E)$ for every subset $E$ ;
$E$ is Lebesgue measurable if and only if $E + h$ is;
$λ_{n} (E + h) = λ_{n} (E)$ for every Lebesgue measurable $E$ .

No choice principle is used. Lebesgue outer measure is defined as an infimum and the Carathéodory condition is a family of equations between its values, so all three clauses are statements about objects that exist in ZF; countable choice is needed to know that $λ_{n}$ is a measure, not to know that it is translation invariant.

Facts & Assumptions

Given: A natural number $n \geq 1$ , a vector $h \in R^{n}$ , and a subset $E \subseteq R^{n}$ .

[L1]

$λ_{n}^{*} (E) : = \inf {\sum_{k = 0}^{\infty} μ_{0} (A_{k}) : A_{k} \in E_{n}$ for every $k$ and $E \subseteq ⋃_{k} A_{k}}$ (Lebesgue outer measure on $R^{n}$ , Series in the nonnegative extended real line).

[L2]

$B (a, b) : = {x \in R^{n} : a_{i} < x_{i} \leq b_{i} for every i < n}$ ; a box is nonempty exactly when $a_{i} < b_{i}$ for every $i < n$ ; and for a nonempty box with real parameters $vol (B) : = \prod_{i < n} (b_{i} - a_{i})$ , the value being $+ \infty$ when a parameter is infinite (Half-open boxes in $R^{n}$ and their volume).

[L3]

A subset $E \subseteq R^{n}$ is an elementary set when there are a natural number $m$ and a list $B_{0}, \dots, B_{m - 1}$ of half-open boxes with $E = ⋃_{j < m} B_{j}$ (Elementary sets: the finite unions of half-open boxes in $R^{n}$ ), and every elementary set is the union of a finite list of pairwise disjoint half-open boxes (Every elementary set is a finite disjoint union of half-open boxes, and any finitely many boxes admit a common grid refinement).

[L4]

For every $n \geq 1$ , elementary volume $μ_{0}$ on $E_{n}$ has value at $A$ the sum of the volumes of the members of any presentation of $A$ by a finite list of pairwise disjoint half-open boxes (The sum of the volumes of a disjoint box decomposition of an elementary set does not depend on the decomposition).

[L5]

A set $E$ is Lebesgue measurable when $λ_{n}^{*} (A) = λ_{n}^{*} (A \cap E) + λ_{n}^{*} (A ∖ E)$ for every $A \subseteq R^{n}$ , and $λ_{n}$ is the restriction of $λ_{n}^{*}$ to the family of these (Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ , Carathéodory measurable sets).

[F1]

The translate of $E \subseteq R^{n}$ by $a$ is $E + a : = {x + a : x \in E}$ ; translation by $a$ is the bijection $τ_{a} (x) = x + a$ , whose inverse is $τ_{- a}$ (Translation of a subset of $R^{n}$ ).

[F2]

Addition of a real to an extended real is defined in every case, with $a + b : = + \infty$ when $a = + \infty$ and $b \neq - \infty$ , and $a + b : = - \infty$ when $a = - \infty$ and $b \neq + \infty$ (The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined).

Proof

technique · direct

1.1L2F1F2

For a parameter pair $(a, b)$ one has $B (a, b) + h = B (a + h, b + h)$ , where $a + h$ is the parameter $i \mapsto a_{i} + h_{i}$ : a point $y$ lies in the left side exactly when $y - h$ satisfies $a_{i} < y_{i} - h_{i} \leq b_{i}$ , that is $a_{i} + h_{i} < y_{i} \leq b_{i} + h_{i}$ . The translated box is empty exactly when the original is, and has the same volume, because $(b_{i} + h_{i}) - (a_{i} + h_{i}) = b_{i} - a_{i}$ when both are real and an infinite parameter stays infinite.

2.1step 1.1L3L4

Consequently, if $A = ⋃_{j < q} B_{j}$ is a presentation of an elementary set by pairwise disjoint half-open boxes, then $A + h = ⋃_{j < q} (B_{j} + h)$ is such a presentation of $A + h$ , so $A + h$ is elementary and $μ_{0} (A + h) = μ_{0} (A)$ .

3.1step 2.1L1F1

A sequence $(A_{k})$ of elementary sets covers $E$ if and only if the sequence $(A_{k} + h)$ covers $E + h$ , and the two covering costs are equal by step 2.1; the correspondence is a bijection between the two families of covers, with inverse given by translating by $- h$ , so the two infima agree and $λ_{n}^{*} (E + h) = λ_{n}^{*} (E)$ .

4.1step 3.1L5F1∎

For test sets, $A \cap (E + h) = ((A - h) \cap E) + h$ and $A ∖ (E + h) = ((A - h) ∖ E) + h$ , so by step 3.1 the Carathéodory identity for $E + h$ tested against $A$ is exactly the identity for $E$ tested against $A - h$ ; as $A$ ranges over all subsets so does $A - h$ , and therefore $E + h$ is Lebesgue measurable if and only if $E$ is, with $λ_{n} (E + h) = λ_{n}^{*} (E + h) = λ_{n}^{*} (E) = λ_{n} (E)$ in that case.

Depends on

Used by

The constructed classical solutions are locally determined by the Cauchy data Corollary
The Nyquist no-aliasing condition Corollary
The Solovay model has no Banach–Tarski decomposition Corollary
The Solovay model has no Vitali or Bernstein set Corollary
Assuming Choice, a proper subgroup of (ℝ,+) can be nonmeasurable Counterexample
Outward dilation defeats subcritical inclusion and homogeneous Poincare on Euclidean space Counterexample
Poincare-Wirtinger fails on disconnected bounded domains Counterexample
Rellich compactness fails on ℝⁿ by translations Counterexample
The regular representation of R is not a Hilbert direct sum of irreducibles Counterexample
The translation semigroup is not analytic Counterexample
Difference quotients on a shrunken domain Definition
Period-one Fourier coefficients, partial sums, and convolution on the torus Definition
Spherical means and the weighted ball integral of space-dependent data Definition
The heat evolution Hₜ of initial data Definition
The one-dimensional torus and its normalized Haar integral Definition
A point source produces a uniform expanding sphere Example
Affine flux reduces the entropy semigroup to translation Example
Conserved energy of a travelling wave packet Example
Følner sets in ℝⁿ Example
Heat evolution of affine and quadratic polynomials Example
Lebesgue measure as Haar measure on rn Example
Mollifying 1_[0,1] at two scales Example
Normalized Haar measure on a torus Example
Steinhaus follows in two lines from the density theorem Example
The Aₚ range of a power weight Example
The right-translation semigroup on Lp has the weak derivative as generator Example
The volume contradiction for an alleged Banach–Tarski decomposition Example
False: measure-preserving transformations are invertible False statement
False: Poincare recurrence needs no finite total measure False statement
A Lebesgue measurable subset of [0,1] that is invariant under changing finitely many binary digits has measure 0 or 1 Lemma
A shear sends the unit cube to a set of Lebesgue measure one Lemma
A translation-invariant L1 function on the line is zero Lemma
Ball and cube maximal functions are pointwise comparable Lemma
Ball means and sphere means are related by a radial derivative Lemma
Blichfeldt lattice-point principle Lemma
Calderon reproducing pair and the telescoping identity in S' Lemma
Chacon partial maps extend to an invertible map mod null sets Lemma
Compactly supported scaled Euclidean bumps Lemma
Complex translation, convolution, approximate identities, and mollification Lemma
Continuous compactly supported functions are translation-continuous in Lᵖ Lemma

…and 47 more results.

Dependency tree · two levels

37 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), Proposition 2.16 (standard reference, not scraped)
E. A. Carlen, Notes on Lebesgue Measure on $\mathbb{R}^n$ and $S^{n-1}$ (Rutgers Math 501), Theorem 2.3 (standard reference, not scraped)