Alphabeta MathHausdorff Measure and Hausdorff Dimension
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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
- 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
- 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
Definition
Extended diameter for Hausdorff covers
Definition
Hausdorff content at a prescribed scale
Lemma
The small-scale Hausdorff limit exists
Definition
Unnormalised Hausdorff measure
Theorem
Hausdorff measure is an outer measure
Theorem
Hausdorff measure is metric and measures every Borel set
Proposition
Zero-dimensional Hausdorff measure is counting measure
Theorem
Hausdorff measure is Borel regular
Proposition
Borel cores of sigma-finite Hausdorff measurable sets
Proposition
Content and measure have the same null sets
Theorem
Lipschitz maps control Hausdorff measure
Corollary
Similarities scale Hausdorff measure exactly
Lemma
Increasing the exponent past finite measure gives zero
Definition
Hausdorff dimension
Theorem
Hausdorff dimension is the unique critical exponent
Theorem
Hausdorff dimension is monotone and countably stable
Corollary
Lipschitz monotonicity and bi-Lipschitz invariance of dimension
Lemma
Elementary lower and upper bounds on a unit cube
Theorem
One-dimensional Hausdorff measure on the line is Lebesgue outer measure
Theorem
Euclidean Hausdorff measure is proportional to Lebesgue measure
Corollary
Euclidean space and positive-volume sets have their Euclidean dimension
Theorem
The mass distribution principle
Lemma
Cantor basic intervals have their expected masses
Lemma
The sharp interval bound for Cantor measure
Theorem
The Cantor set has dimension log 2 / log 3 and critical measure one
Definition
Sets defined by permitted binary digit positions
Proposition
Digit-position density determines Hausdorff dimension
Remark
Dimension leaves the critical measure undetermined
Remark
Chart surface measure and Hausdorff measure: a boundary of scope
Remark
Orientation for the published Weierstrass graph remark
Sources
- Fremlin 264A, 264K
- Bishop–Peres, Fractals in Probability and Analysis, §1.2 pp.4–6
- Fremlin, Measure Theory, 264A,D(b),K
- Fremlin 264D(d),264K
- Fremlin 264A–C,K
- Fremlin 264B and 264Xa
- Fremlin, Measure Theory, 264C,E
- Bishop–Peres, Theorem 1.2.4
- Fremlin 264D(b),264G(a)
- Fremlin, Measure Theory, 264F(a,b),264K,264Xd,264Ye
- Falconer, The Geometry of Fractal Sets, Theorem 1.6(a)
- Fremlin, Measure Theory, 264F(c)
- Falconer, The Geometry of Fractal Sets, Theorem 1.6(b)
- Bishop–Peres Proposition 1.2.6; Fremlin 264Xa
- Fremlin, Measure Theory, 264G,264Yj(i)
- Falconer, The Geometry of Fractal Sets, Lemma 1.8
- Fremlin 264G,264Yf(i),264Yj(i)
- Bishop–Peres Lemma 1.2.5; Fremlin 264Xe
- Bishop–Peres Definition 1.2.1 and Proposition 1.2.6
- Bishop–Peres Proposition 1.2.6; Fremlin 264Yk
- Bishop–Peres §1.1 closing paragraph (p.3), Example 1.2.7, Exercise 1.6
- Fremlin 264G,Yj(i),Yk; Falconer Lemma 1.8
- Falconer §1.2 p.8 cube upper estimate; §1.4 pp.12–13 volume covers; Fremlin 264H(b)
- Falconer §1.4 p.12, immediately before Theorem 1.11
- Fremlin 264I statement (weaker constant); design MT-21 prescribed uniqueness route
- Bishop–Peres §1.2 p.7 after Lemma 1.2.8; Falconer §1.2 p.8
- Bishop–Peres Lemma 1.2.8
- Fremlin 264J(a,d); Bishop–Peres Example 1.4.2 (uniform digits)
- Fremlin, Measure Theory, 264J(c,d), adapted from cylinder counts to the existing Cantor measure
- Fremlin, Measure Theory, 264J
- Hunter, Measure Theory, Theorem 2.34 and Example 2.37
- Bishop–Peres Example 1.3.2 and Example 1.4.2
- Bishop–Peres Examples 1.3.2,1.4.2; §1.3 grid comparison
- Bishop–Peres Proposition 1.2.6 and Example 1.4.2
- Fremlin chapter 26 introduction and 265A opening; design geometric-measure receipt
- Bishop–Peres §1.2 p.9 Weierstrass discussion; design forward-reference receipt