View source for Measure-preserving transformation

''of a [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m0632701.png" />.''

A [[Measurable mapping|measurable mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m0632702.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m0632703.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m0632704.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m0632705.png" /> is called an invariant measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m0632706.png" />. A measurable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m0632707.png" /> between measure spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m0632708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m0632709.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m06327010.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m06327011.png" /> is usually called a measure-preserving mapping. A surjective measure-preserving transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m06327012.png" /> of a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m06327013.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m06327014.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m06327015.png" /> onto itself, is often called an endomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m06327016.png" />; an endomorphism which is bijective and whose inverse is also measure preserving is called an [[Automorphism|automorphism]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063270/m06327017.png" />.

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]].