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
.