invariant by a measure-preserving transformation


invariant \PMlinkescapephraseproperties \PMlinkescapephraseproperty

Let X be a set and T:XX a transformation of X.

The notion of invariance by T we are about to describe is stronger than the usual notion of invariance (, and is especially useful in ergodic theory. Thus, in most applications, (X,𝔅,μ) is a measure spaceMathworldPlanetmath and T is a measure-preserving transformationPlanetmathPlanetmath. Nevertheless, the definition of invariance and its properties are general and do not require any such assumptions.

Definition - A subset AX is said to be invariant by T, or T-invariant, if T-1(A)=A.

The fundamental property of this concept is the following: if A is invariant by T, then so is XA.

Thus, when A is invariant by T we obtain by restrictionPlanetmathPlanetmath two well-defined transformations


Hence, the existence of an allows one to decompose the set X into two disjoint subsets and study the transformation T in each of these subsets.

Remark - When T is a measure-preserving transformation in a measure space (X,𝔅,μ) one usually restricts the notion of invariance to measurable subsets A𝔅.

Title invariant by a measure-preserving transformation
Canonical name InvariantByAMeasurepreservingTransformation
Date of creation 2013-03-22 18:04:15
Last modified on 2013-03-22 18:04:15
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 4
Author asteroid (17536)
Entry type Definition
Classification msc 03E20
Classification msc 28D05
Classification msc 37A05