# invariant by a measure-preserving transformation

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invariant \PMlinkescapephraseproperties \PMlinkescapephraseproperty

Let $X$ be a set and $T:X\u27f6X$ a transformation of $X$.

The notion of invariance by $T$ we are about to describe is stronger than the usual notion of invariance (http://planetmath.org/invariant), and is especially useful in ergodic theory. Thus, in most applications, $(X,\U0001d505,\mu )$ is a measure space^{} and $T$ is a measure-preserving transformation^{}. Nevertheless, the definition of invariance and its properties are general and do not require any such assumptions.

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Definition - A subset $A\subseteq X$ is said to be *invariant* by $T$, or $T$-*invariant*, if ${T}^{-1}(A)=A$.

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The fundamental property of this concept is the following: if $A$ is invariant by $T$, then so is $X\setminus A$.

Thus, when $A$ is invariant by $T$ we obtain by restriction^{} two well-defined transformations

${T|}_{A}:A\u27f6A$ | ||

${T|}_{X\setminus A}:X\setminus A\u27f6X\setminus A$ |

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 $(X,\U0001d505,\mu )$ one usually restricts the notion of invariance to measurable subsets $A\in \U0001d505$.

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 |