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 $\mu(T^{-1}(\{0\}))=\mu(\mathbb{R})=\infty$ ).

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