Alphabeta MathHausdorff Measure and Hausdorff Dimension — Examples
About this page
Pipeline-generated
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.
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Variation and the Riemann–Stieltjes Integral
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Hausdorff Measure and Hausdorff Dimension
- Lebesgue Measure on Euclidean Space
- Lebesgue-Stieltjes Measures and Distribution Functions
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Non Measurable Sets and the Cost of Choice
- Order, Zorn's Lemma, and the Axiom of Choice
- Outer Measure and the Caratheodory Extension Theorem
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Subspaces, Products, and Quotients
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Example
The ordinary Cantor set at its critical exponent
Example
The fat Cantor set has positive length and dimension one
Example
A planar segment has Hausdorff measure equal to length
Example
A Lipschitz graph has finite Hausdorff length
Example
The rationals are dense but have dimension zero
Example
A Sierpinski gasket computed by hand
Counterexample
A dimension-one set can have zero length
Counterexample
An uncountable compact set can have dimension zero
Counterexample
A continuous image can raise Hausdorff dimension
False statement
Hausdorff measure is countably additive on every subset
False statement
Dimensions add under unions
False statement
Critical Hausdorff measure is always finite and positive
False statement
Continuous injections preserve Hausdorff dimension
False statement
Dimension zero forces countability
False statement
Vanishing at all positive exponents forces countability
Sources
- Fremlin 264J
- Bishop–Peres §1.2 p.7 positive-volume consequence; existing fat Cantor construction
- Fremlin 264G; Falconer §1.4 p.12
- Fremlin 264G and 264Xf(i) (specialised Lipschitz graph)
- Bishop–Peres Example 1.2.7 (countable nullity), specialised to Q
- Bishop–Peres Example 1.3.4 pp.15–16
- Bishop–Peres Example 1.4.2, nonsquare-position specialisation
- Bishop–Peres Example 1.4.2, square-position specialisation
- Fremlin 264J; published Cantor-function ternary/binary formula
- Fremlin 264C,I; existing Vitali theorem
- Bishop–Peres §1.1 p.3 countable stability; Proposition 1.2.6
- Bishop–Peres Proposition 1.2.6 and Example 1.4.2
- Semmes §2.5 pp.31–32, snowflake metric and Hausdorff measure identity