Measure-preserving transformation

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

2010 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.

How to Cite This Entry:
Measure-preserving transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measure-preserving_transformation&oldid=47815