LemmaStatement: 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.

The small-scale Hausdorff limit exists

Statement

For $A \subseteq B \subseteq X$ and $0 < η \leq δ \leq \infty$ ,

$H_{δ}^{s} (A) \leq H_{δ}^{s} (B), H_{δ}^{s} (A) \leq H_{η}^{s} (A) .$

For every metric space, every subset $A$ , and every finite $s \geq 0$ ,

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

No separability or existence of a countable small-scale cover is assumed.

Facts & Assumptions

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

[F1]

The scale value is the infimum of admissible covering costs, with empty infimum $\infty$ . Hausdorff content at a prescribed scale

[F2]

Every subset of the extended real line has a supremum and infimum in that ordered set. 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 $R$

Proof

1.1F1

Every cover of $B$ covers $A$ , and every $η$ -cover is a $δ$ -cover. Infima over the larger families are smaller, including when a family is empty.

2.1F2step 1.1

The dyadic values form a nondecreasing nonnegative sequence. Its supremum $M$ exists; if finite, the definition of supremum gives eventual values above $M - ε$ , and if infinite, eventual values above every real bound. Thus the sequence has extended limit $M$ .

3.1F1step 1.1step 2.1∎

For each $δ > 0$ there is $k$ with $2^{- k} \leq δ$ , so $H_{δ}^{s} (A) \leq M$ . Conversely every dyadic value occurs among the scale values. Their suprema agree. For $A = \emptyset$ all values are zero; no part divided by $s$ , so $s = 0$ is included.

Depends on

Used by

Unnormalised Hausdorff measure Definition

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 264D(d),264K (standard reference, not scraped)