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

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Lipschitz maps control Hausdorff measure

Statement

Assume the Axiom of Countable Choice. Let $f : D \subseteq X \to Y$ be an $L$ -Lipschitz map between metric spaces and $A \subseteq D$ . For finite $s \geq 0$ and $L > 0$ ,

$H^{s} (f (A)) \leq L^{s} H^{s} (A) .$

If $L = 0$ , the image is empty or a singleton: $H^{s} (f (A)) = 0$ for $s > 0$ and $H^{0} (f (A)) \leq H^{0} (A)$ . No product $0 \cdot \infty$ is used.

Facts & Assumptions

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

[F1]

Hausdorff outer values come from arbitrary nonempty covers and small-scale suprema. Unnormalised Hausdorff measure

[F2]

Under the standing Countable Choice hypothesis, $H^{0}$ is counting measure. Zero-dimensional Hausdorff measure is counting measure

[F3]

$L$ -Lipschitz means $d_{Y} (f (x), f (y)) \leq L d_{X} (x, y)$ for all points in the domain. Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction

Proof

1.1

For $L > 0$ , replace every member $U$ of a $δ$ -cover of $A$ by $f (U \cap D)$ and discard empty images. The image diameter is at most $L diam U$ . For positive $s$ its cost is at most $L^{s}$ times the old cost; for $s = 0$ each retained member still costs one.

F1F3

2.1

Infimising gives $H_{L δ}^{s} (f (A)) \leq L^{s} H_{δ}^{s} (A)$ ; if the right scale infimum is infinite the inequality is automatic. Let $δ$ tend to zero to obtain the assertion, since $L$ is finite and positive.

F1step 1.1

3.1

For $L = 0$ , any two image points have distance zero, so a nonempty image is a singleton. Its singleton cover costs zero for $s > 0$ . For $s = 0$ , image cardinality cannot exceed domain cardinality, whether finite or infinite. Empty images have zero measure at every exponent.

F1F2F3∎

Depends on

Used by

Lipschitz monotonicity and bi-Lipschitz invariance of dimension Corollary
Similarities scale Hausdorff measure exactly Corollary
A Lipschitz graph has finite Hausdorff length Example

Dependency tree · two levels

13 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, Measure Theory, 264G,264Yj(i) (standard reference, not scraped)
Falconer, The Geometry of Fractal Sets, Lemma 1.8 (standard reference, not scraped)