DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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.

Unnormalised Hausdorff measure

Definition

For every subset $A$ of a metric space and finite real $s \geq 0$ , define the unnormalised Hausdorff measure by

$H^{s} (A) = \sup_{0 < δ < \infty} H_{δ}^{s} (A) = \lim_{k \to \infty} H_{2^{- k}}^{s} (A) .$

The scale functions use Hausdorff content at a prescribed scale and the limit exists by The small-scale Hausdorff limit exists. The name here initially denotes a set function on all subsets. Its outer-measure axioms and the measure on its Carathéodory measurable domain are established below; no measurability of an arbitrary subset is built into the notation. No volume normalising factor is inserted.

Depends on

Used by

Hausdorff dimension Definition
Elementary lower and upper bounds on a unit cube Lemma
Increasing the exponent past finite measure gives zero Lemma
Content and measure have the same null sets Proposition
Zero-dimensional Hausdorff measure is counting measure Proposition
Chart surface measure and Hausdorff measure: a boundary of scope Remark
Hausdorff measure is an outer measure Theorem
Lipschitz maps control Hausdorff measure Theorem
One-dimensional Hausdorff measure on the line is Lebesgue outer measure Theorem
The mass distribution principle Theorem

Dependency tree · two levels

6 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 264A–C,K (standard reference, not scraped)