Definition 1.

Let (E,) be a measurable spaceMathworldPlanetmathPlanetmath, and T:EE be a measurable map. A measureMathworldPlanetmath μ on (E,) is said to be quasi-invariant under T if μT-1 is absolutely continuousMathworldPlanetmath with respect to μ. That is, for all A with μ(A)=0, we also have μ(T-1(A))=0. We also say that T leaves μ quasi-invariant.

As a example, let E= with the Borel σ-algebra (http://planetmath.org/BorelSigmaAlgebra), and μ be Lebesgue measureMathworldPlanetmath. If T(x)=x+5, then μ is quasi-invariant under T. If S(x)=0, then μ is not quasi-invariant under S. (We have μ({0})=0, but μ(T-1({0}))=μ()=).

To give another example, take E to be the nonnegative integers and declare every subset of E to be a measurable set. Fix λ>0. Let μ({n})=λnn! and extend μ to all subsets by additivity. Let T be the shift function: nn+1. Then μ 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