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.

Hausdorff dimension

Definition

For a subset $A$ of a metric space define

$\dim_{H} A = \inf {s \in [0, \infty) : H^{s} (A) = 0} \in [0, \infty] .$

Here $H^{s}$ is Unnormalised Hausdorff measure, $s$ always denotes a finite real exponent, and $\inf \emptyset = \infty$ , in the extended order of Every subset of $\overline{R}$ has a least upper bound and a greatest lower bound in $\overline{R}$ , agreeing with the real supremum and infimum on nonempty sets bounded in $R$ . Thus $\dim_{H} \emptyset = 0$ . Dimension itself may be infinite, but no measure with exponent $\infty$ is being defined. Any empty supremum over exponents below is taken in $[0, \infty]$ and equals zero, not the empty supremum in the whole extended real line.

Depends on

Used by

Dependency tree · two levels

12 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

Bishop–Peres Definition 1.2.1 and Proposition 1.2.6 (standard reference, not scraped)