Measure-preserving transformation

of a measure space $ ( X , \mathfrak A , \mu ) $.

2020 Mathematics Subject Classification: Primary: 28D05 [MSN][ZBL]

A measurable mapping $ T : X \rightarrow X $ such that $ \mu ( T ^ {-} 1 ( A) ) = \nu ( A) $ for every $ A \in \mathfrak A $; $ \mu $ is called an invariant measure for $ T $. A measurable mapping $ T : X \rightarrow Y $ between measure spaces $ ( X , \mathfrak A , \mu ) $ and $ ( Y , \mathfrak B , \nu ) $ such that $ \mu ( T ^ {-} 1 ( B) ) = \nu ( B) $ for every $ B \in \mathfrak B $ is usually called a measure-preserving mapping. A surjective measure-preserving transformation $ T $ of a measure space $ ( X , \mathfrak A , \mu ) $, i.e., $ T $ maps $ X $ onto itself, is often called an endomorphism of $ ( X , \mathfrak A , \mu ) $; an endomorphism which is bijective and whose inverse is also measure preserving is called an automorphism of $ ( X , \mathfrak A , \mu ) $.

Measure-preserving transformations arise, for example, in the study of classical dynamical systems (cf. (measurable) Cascade; Measurable flow). In that case the transformation is first obtained as a continuous (or smooth) transformation of some, often compact, topological space (or manifold), and the existence of an invariant measure is proved. An example is Liouville's theorem for a Hamiltonian system (cf. also Liouville theorems).

For further information and references see Ergodic theory.