Alphabeta MathMeasure-Preserving Systems and Mixing Criteria
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- 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
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- 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 Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Invertible measure-preserving systems
Theorem
Measure preservation can be checked on a generating pi-system
Proposition
Compositions, iterates and completions preserve invariance
Definition
The Koopman operator
Theorem
Koopman operators are linear isometries
Definition
Strict and mod-null invariant sigma-algebras
Proposition
Both invariant families are sigma-algebras
Lemma
Mod-null invariant sets have strict representatives
Definition
Ergodicity relative to an invariant measure
Theorem
Equivalent invariant-set and invariant-function criteria for ergodicity
Proposition
Positive sets sweep out ergodic probability systems
Definition
Strong and weak mixing on a probability space
Theorem
Mixing implies weak mixing, which implies ergodicity
Lemma
Approximation in symmetric difference by a generating algebra
Theorem
Mixing is checkable on a generating pi-system
Proposition
Mixing correlations extend to L2 functions
Sources
- Einsiedler–Ward Definition 2.1 p.13 and conull restriction convention in Definition 2.7 p.16
- E–W §2.1 p.13; local sigma-finite uniqueness theorem
- Einsiedler–Ward Exercise 2.1.3, p.19; Sarig Proposition 1.4 preservation proof, p.8
- E–W §2.4 pp.28–29; Sarig Proposition 1.3
- Einsiedler–Ward §2.4 opening, pp.28–29 (isometry paragraph, not Lemma 2.18)
- E–W Proposition 2.14; Sarig Proposition 1.1
- Sarig Proposition 1.1 proof; E–W Proposition 2.14
- E–W Proposition 2.14 pp.23–24; Sarig Proposition 1.1 pp.5–6
- Sarig Definition 1.4 p.5
- E–W Proposition 2.14 pp.23–25; Sarig Proposition 1.1
- E–W Proposition 2.14 pp.24–25
- E–W Definitions 2.32 and 2.35, pp.49–50
- E–W §2.7 pp.49–50; Sarig Proposition 1.2
- E–W Proposition 2.15 proof and Exercise 2.7.3
- Einsiedler–Ward Exercise 2.7.3(1)–(2), pp.52–53; local Boolean-algebra extension from a pi-system
- Sarig Proposition 1.3 p.7; E–W Exercise 2.7.7