ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.
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The fat Cantor set has positive length and dimension one

Example

Assume the Axiom of Countable Choice. For the Smith–Volterra–Cantor set $S$ ,

$H^{1} (S) = λ_{1} (S) = \frac{1}{2}, \dim_{H} S = 1.$

Facts & Assumptions

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

[F1]

The stage- $n$ set $S_{n}$ consists of $2^{n}$ intervals of length $ℓ_{n}$ , with $ℓ_{0} = 1$ and $ℓ_{n + 1} = (ℓ_{n} - 4^{- n - 1}) / 2$ ; the stages decrease to $S$ . The Smith-Volterra-Cantor set: the same construction removing, at stage $n \geq 1$ , an open middle interval of length $4^{- n}$ from each of the $2^{n - 1}$ remaining intervals

[F2]

The fat Cantor set is closed and bounded; every interval cover has total length at least $1 / 2$ . The Smith-Volterra-Cantor set is compact, perfect and nowhere dense, and does not have measure zero

[F3]

Under the standing Countable Choice hypothesis, on the line $H^{1}$ equals Lebesgue outer measure. One-dimensional Hausdorff measure on the line is Lebesgue outer measure

[F4]

Finite positive measure at exponent one forces dimension one. Hausdorff dimension is the unique critical exponent

[F5]

For decreasing measurable sets, continuity from above holds if some member has finite measure. Continuity from above when one set has finite measure

Verification

1.1

The construction intervals at a fixed level are disjoint, and induction in the defining recursion gives $2^{n} ℓ_{n} = \frac{1}{2} + 2^{- n - 1}$ . Thus $λ_{1} (S_{n}) = \frac{1}{2} + 2^{- n - 1}$ . The stages are closed, and $λ_{1} (S_{0}) = 1 < \infty$ . Their intersection is the closed set $S$ .

F1F2

2.1

Continuity from above yields $λ_{1} (S) = \lim_{n} λ_{1} (S_{n}) = 1 / 2$ . The equality with $H^{1}$ and the finite-positive criterion give the stated measure and dimension. Thus the earlier cover lower bound has been matched by an exact measure calculation here.

F3F4F5step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

49 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 §1.2 p.7 positive-volume consequence; existing fat Cantor construction (standard reference, not scraped)