TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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.

Jordan inner content, outer content, measurability, and content are translation invariant

Statement

Let $n \geq 1$ . For every bounded $E \subseteq R^{n}$ and every $a \in R^{n}$ , the translates $E + a$ and $E$ have equal Jordan inner and outer contents. Consequently, $E$ is Jordan measurable if and only if $E + a$ is Jordan measurable, and in that case

$cont (E + a) = cont (E) .$

Here translation is as in Translation of a subset of $R^{n}$ .

Facts & Assumptions

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

[L1]

For a bounded set, Jordan outer content is the infimum of the total volumes of finite axis-parallel rectangle covers, and Jordan inner content is the supremum of the total volumes of finite interior-disjoint axis-parallel rectangle families contained in the set (Jordan inner and outer content and Jordan measurable bounded sets in $R^{m}$ ).

[L2]

A rectangle $[u, v] \subseteq R^{n}$ has volume $\prod_{j < n} (v_{j} - u_{j})$ (Axis-parallel rectangles in $R^{m}$ and their volume).

[L3]

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

Proof

technique · direct

1.1

Translation by $a$ sends every axis-parallel rectangle $[u, v]$ bijectively to $[u + a, v + a]$ , preserves all side lengths and volumes by [L2], preserves containment and interior-disjointness, and sends finite covers or inner families for $E$ to families of the same total volume for $E + a$ ; this also covers the empty family, the empty set, and rectangles with zero side length.

L1L2L3

2.1

Step 1.1 gives outer content of $E + a$ at most that of $E$ and inner content of $E + a$ at least that of $E$ ; applying the same argument to translation by $- a$ gives the reverse inequalities, so both respective contents are equal.

step 1.1L1

3.1

Equality of the two contents for $E$ is therefore equivalent to equality of the two contents for $E + a$ , and when these equalities hold their common values agree.

step 2.1L1∎

Depends on

Used by

Dependency tree · two levels

19 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

M. E. Taylor, Introduction to Analysis in Several Variables, §3.1 (standard reference, not scraped)
W. F. Trench, Introduction to Real Analysis, §7.3 (standard reference, not scraped)