Probability, Random Processes, and Ergodic Properties

Introduction.- Probability Spaces.- Sample Spaces.- Metric Spaces.- Measurable Spaces.- Borel Measurable Spaces.- Polish Spaces.- Probability Spaces.- Complete Probability Spaces.- Extension.- Random Processes and Dynamical Systems.- Measurable Functions and Random Variables.- Approximation of Random Variables and Distributions.- Random Processes and Dynamical Systems.- Distributions.- Equivalent Random Processes.- Codes, Filters, and Factors.- Isomorphism.- Standard Alphabets.- Extension of Probability Measures.- Standard Spaces.- Some Properties of Standard Spaces.- Simple Standard Spaces.- Characterization of Standard Spaces.- Extension in Standard Spaces.- The Kolmogorov Extension Theorem.- Bernoulli Processes.- Discrete B-Processes.- Extension Without a Basis.- Lebesgue Spaces.- Lebesgue Measure on the Real Line.- Standard Borel Spaces.- Products of Polish Spaces.- Subspaces of Polish Spaces.- Polish Schemes.- Product Measures.- IID Random Processes and B-processes.- Standard Spaces vs. Lebesgue Spaces.- Averages.- Discrete Measurements.- Quantization.- Expectation.- Limits.- Inequalities.- Integrating to the Limit.- Time Averages.- Convergence of Random Variables.- Stationary Random Processes.- Block and Asymptotic Stationarity.- Conditional Probability and Expectation.- Measurements and Events.- Restrictions of Measures.- Elementary Conditional Probability.- Projections.- The Radon-Nikodym Theorem.- Probability Densities.- Conditional Probability.- Regular Conditional Probability.- Conditional Expectation.- Independence and Markov Chains.- Ergodic Properties.- Ergodic Properties of Dynamical Systems.- Implications of Ergodic Properties.- Asymptotically Mean Stationary Processes.- Recurrence.- Asymptotic Mean Expectations.- Limiting Sample Averages.- Ergodicity.- Block Ergodic and Totally Ergodic Processes.- The Ergodic Decomposition.- Ergodic Theorems.- The Pointwise Ergodic Theorem.- Mixing Random Processes.- Block AMS Processes.- The Ergodic Decomposition of AMS Systems.- The Subadditive Ergodic Theorem.- Process Approximation and Metrics.- Distributional Distance.- Optimal Coupling Distortion and dp Distance.- Prohorov and Variational Distances.- Evaluating dp .- Measures on Measures.- The Ergodic Decomposition.- The Ergodic Decomposition Revisited.- The Ergodic Decomposition of Markov Processes.- Barycenters.- Affine Functions of Measures.- The Ergodic Decomposition of Affine Functionals.- References.-