# quasi-invariant

###### Definition 1.

Let $(E,\mathcal{B})$ be a measurable space^{}, and $T:E\to E$ be a measurable map. A measure^{} $\mu $ on $(E,\mathcal{B})$ is said to be *quasi-invariant* under $T$ if $\mu \circ {T}^{-1}$ is absolutely continuous^{} with respect to $\mu $. That is, for all $A\in \mathcal{B}$ with $\mu (A)=0$, we also have $\mu ({T}^{-1}(A))=0$. We also say that $T$ leaves $\mu $ quasi-invariant.

As a example, let $E=\mathbb{R}$ with $\mathcal{B}$ the Borel $\sigma $-algebra (http://planetmath.org/BorelSigmaAlgebra), and $\mu $ be Lebesgue measure^{}. If $T(x)=x+5$, then $\mu $ is quasi-invariant under $T$. If $S(x)=0$, then $\mu $ is not quasi-invariant under $S$. (We have $\mu (\{0\})=0$, but $\mu ({T}^{-1}(\{0\}))=\mu (\mathbb{R})=\mathrm{\infty}$).

To give another example, take $E$ to be the nonnegative integers and declare every subset of $E$ to be a measurable set. Fix $\lambda >0$. Let $\mu (\{n\})=\frac{{\lambda}^{n}}{n!}$ and extend $\mu $ to all subsets by additivity. Let $T$ be the shift function: $n\to n+1$. Then $\mu $ is quasi-invariant under $T$ and not invariant (http://planetmath.org/HaarMeasure).

Title | quasi-invariant |
---|---|

Canonical name | Quasiinvariant |

Date of creation | 2013-03-22 15:56:00 |

Last modified on | 2013-03-22 15:56:00 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 12 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 28A12 |

Related topic | RepresentationsOfLocallyCompactGroupoids |