Measure-preserving dynamical system

In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory.

It is defined as a probability space and a measure-preserving transformation on it. In more detail, it is system with the following structure:

is a set,

is a -algebra over ,

is a probability measure, so that , and

is a measurable transformation which preserves the measure , i. e. each measurable satisfies .

For example m could be the normalised angle measure dθ/2π on the unit circle, and T a rotation.