View source for Measure-preserving transformation

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''of a [[Measure space|measure space]]  $  ( X , \mathfrak A , \mu ) $.''

{{MSC|28D05}}

[[Category:Measure-theoretic ergodic theory]]

A [[Measurable mapping|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|automorphism]] of  $  ( X , \mathfrak A , \mu ) $.

Measure-preserving transformations arise, for example, in the study of classical dynamical systems (cf. (measurable) [[Cascade|Cascade]]; [[Measurable flow|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|Hamiltonian system]] (cf. also [[Liouville theorems|Liouville theorems]]).

For further information and references see [[Ergodic theory|Ergodic theory]].