Alphabeta MathThe Ergodic Theorems of von Neumann and Birkhoff
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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
- 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
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
- Measure Preserving Transformations and Poincare Recurrence
- Measure-Preserving Systems and Mixing Criteria
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
- Weak Mixing and the Chacon Transformation
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- 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
Ergodic partial sums, time averages, and the invariant L2 subspace
Proposition
Ergodic averages are measurable, representative independent, and Lp contractive
Theorem
Maximal ergodic theorem
Lemma
Sigma-finite ergodic oscillation sets have finite measure
Theorem
Birkhoff pointwise ergodic theorem
Theorem
Finite-measure identification of the Birkhoff limit
Lemma
Ergodic averages converge in Lp on finite-measure spaces
Corollary
Birkhoff ergodic theorem for ergodic finite-measure systems
Theorem
Von Neumann mean ergodic theorem in L2
Definition
Unique ergodicity
Lemma
Continuous functions determine Borel probabilities on compact metric spaces
Theorem
Unique ergodicity is equivalent to uniform ergodic averages
Lemma
The unit-interval circle is a nonempty compact metric space
Theorem
Irrational circle rotations are uniquely ergodic
Definition
Equidistribution modulo one
Theorem
Weyl equidistribution for irrational rotations
Definition
Canonical base-b expansions and normal numbers
Lemma
Base-b digit cylinders are orbit cylinders
Theorem
Borel's normal number theorem
Theorem
Fair-coin frequency strong law
False statement
Birkhoff averages need not converge at every point
False statement
Ergodic averages need not equal the space mean without ergodicity
False statement
A Birkhoff limit need not be constant
False statement
Norm convergence alone does not imply pointwise convergence
False statement
Weyl equidistribution fails for some rotation angles
False statement
Not every real number is normal
False statement