LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04

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.

Euclidean balls have positive finite Lebesgue measure

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Let $x \in R^{n}$ and let $r > 0$ . Then the Euclidean ball $B (x, r)$ is Lebesgue measurable and satisfies $0 < λ (B (x, r)) < \infty .$

Facts & Assumptions

Given: The Axiom of Countable Choice, a point $x \in R^{n}$ , and a real radius $r > 0$ .

[L1]

The Euclidean ball is $B (x, r) = {y \in R^{n} : ∥ y - x ∥_{2} < r}$ . (Open ball, closed ball and sphere in a metric space)

[L2]

Every box between its open and closed forms is Lebesgue measurable with its usual volume. (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)

Proof

technique · direct

1.1

If $∥ y - x ∥_{\infty} < r / \sqrt{n}$ , then [L1, algebra] $∥ y - x ∥_{2} \leq \sqrt{n} ∥ y - x ∥_{\infty} < r,$ so the open cube $x + (- r / \sqrt{n}, r / \sqrt{n})^{n}$ is contained in $B (x, r)$ . On the other hand, $∥ y - x ∥_{2} < r$ implies $∣ y_{i} - x_{i} ∣ < r$ for every coordinate, so $B (x, r) \subseteq x + (- r, r)^{n}$ .

L1algebra

2.1

The two cubes from step 1.1 are measurable by [L2], with measures [step 1.1, L2, algebra] $(2 r / \sqrt{n})^{n} > 0 and (2 r)^{n} < \infty .$ Since $B (x, r)$ lies between them, monotonicity gives $0 < (2 r / \sqrt{n})^{n} \leq λ (B (x, r)) \leq (2 r)^{n} < \infty .$

step 1.1L2algebra

3.1

Therefore $B (x, r)$ is Lebesgue measurable of positive finite measure. [step 2.1]

step 2.1∎

Depends on

Used by

A family shrinking nicely to a point Definition
Density of a measurable set at a point Definition
The average of a locally integrable function over a Euclidean ball Definition
Ball averages vary continuously with the centre and radius Proposition
Lebesgue density theorem Theorem

Dependency tree · two levels

26 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

Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Lemma 3.16 (standard reference, not scraped)