CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Similarities scale Hausdorff measure exactly

Statement

Assume the Axiom of Countable Choice. If $f : D \subseteq X \to Y$ satisfies $d_{Y} (f (x), f (y)) = c d_{X} (x, y)$ for a finite constant $c > 0$ , then for every $A \subseteq D$ and finite $s \geq 0$ ,

$H^{s} (f (A)) = c^{s} H^{s} (A) .$

This includes isometries and Euclidean translations and dilations. The Hausdorff outer value of a subset of a metric subspace equals its value computed in the ambient metric space.

Facts & Assumptions

Given: The objects, conventions, and hypotheses in the statement above.

[F1]

Under the standing Countable Choice hypothesis, for a positive Lipschitz constant $L$ , $H^{s} (f (A)) \leq L^{s} H^{s} (A)$ , including $s = 0$ . Lipschitz maps control Hausdorff measure

Proof

1.1

Since $c > 0$ , equality of images implies distance zero in $D$ , so $f$ is injective. Its inverse on $f (D)$ is $1 / c$ -Lipschitz. Apply the inequality to $f$ and its inverse to obtain both bounds and hence equality; all scalar multipliers are positive, so infinite values cause no indeterminate product.

2.1

An ambient cover can be intersected with the subspace and its empty members discarded, without increasing cost or diameter. Conversely every subspace cover is an ambient cover. Infima at each scale, and then their suprema, agree. The empty set and $s = 0$ obey the same comparison. Taking $c = 1$ gives isometries and translations; a Euclidean dilation by $c > 0$ scales distances by $c$ .

givenstep 1.1∎

Depends on

Lipschitz maps control Hausdorff measure

Used by

Dependency tree · two levels

4 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

Fremlin 264G,264Yf(i),264Yj(i) (standard reference, not scraped)