Measure-preserving Dynamical System - Definition

Definition

A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system

with the following structure:

• X is a set,
• is a σ-algebra over X,
• is a probability measure, so that μ(X) = 1, and μ(∅) = 0,
• T : X → X is a measurable transformation which preserves the measure μ, i. e. each satisfies μ(T−1A) = μ(A).

This definition can be generalized to the case in which T is not a single transformation that is iterated to give the dynamics of the system, but instead is a monoid (or even a group) of transformations T_s : X → X parametrized by s ∈ Z (or R, or N ∪ {0}, or [0, +∞)), where each transformation T_s satisfies the same requirements as T above. In particular, the transformations obey the rules

• T₀ = id_X : X → X, the identity function on X;
• , whenever all the terms are well-defined;
• , whenever all the terms are well-defined.

The earlier, simpler case fits into this framework by definingT_s = Ts for s ∈ N.

The existence of invariant measures for certain maps and Markov processes is established by the Krylov–Bogolyubov theorem.