Probability spaces and dynamical systems

Core idea

Start with measurable events and probabilities, then add transformations, invariant events, factors, and flows.

Prerequisite doorway

• Set
• Sigma-algebra
• Measurable space
• Measure
• Probability space
• Measurable function
• Null set
• Almost everywhere
• Pushforward measure
• Group Action

Reading path

• Standard Borel space
• Isomorphism of probability spaces modulo null sets
• Standard probability space
• Atom of a measure
• Atomless probability space
• Isomorphism theorem for atomless standard probability spaces
• Measure-preserving transformation
• Probability-preserving dynamical system
• Measure-preserving group action
• Automorphism group of a probability space
• Invariant sigma-algebra
• Ergodic transformation
• Ergodic group action
• Observable of a measurable dynamical system
• Time average of an observable
• Space average of an observable
• Product of probability-preserving systems
• Measurable conjugacy
• Factor of a probability-preserving system
• Measure-preserving flow
• Hilbert space of square-integrable observables
• Invariant probability measure
• Minimal topological dynamical system
• Total ergodicity

Return to Ergodic theory.