1 Definition

Definition - Let (X1,𝔅1,μ1) and (X2,𝔅2,μ2) be measure spacesMathworldPlanetmath, and T:X1X2 be a measurable transformationPlanetmathPlanetmath. The transformation T is said to be measure-preserving if for all A𝔅2 we have that


where T-1(A) is, as usual, the set of points xX1 such that T(x)A.

Additional Notation:

  • Measure-preserving transformations between the same measure space are sometimes called of the measure space.


  • The fact that a map T:X1X2 is measure-preserving depends heavily on the sigma-algebras 𝔅i and measures μi involved. If other measures or sigma-algebras are also in consideration, one should make clear to which measure space is T:X1X2 measure-preserving.

  • Measure-preserving maps are the morphismsMathworldPlanetmathPlanetmath on the category whose objects are measure spaces. This should be clear from the next results and examples.

2 Properties

  • The composition of measure-preserving maps is again measure-preserving. Of course, we are supposing that the domains and codomains of the maps are such that the composition is possible.

  • Let (X1,𝔅1,μ1) and (X2,𝔅2,μ2) be measure spaces and (X1,𝔅1¯,μ1¯) and (X2,𝔅2¯,μ2¯) their completions. If T:(X1,𝔅1,μ1)(X2,𝔅2,μ2) is measure-preserving, then so is T:(X1,𝔅1¯,μ1¯)(X2,𝔅2¯,μ2¯).

  • Let (X1,𝔅1,μ1) and (X2,𝔅2,μ2) be measure spaces and T1:X1X1, T2:X2X2 be measure-preserving maps. Then, the product map T1×T2:X1×X2X1×X2, defined by


    is a measure-preserving transformation of (T1×T2,𝔅1×𝔅1,μ1×μ2).

3 Examples

  • The identity map of a measure space (X,𝔅,μ) is always measure-preserving.

  • Let G be a locally compact group ( For each aG, the transformation T(g):=ag is measure-preserving relatively to any left Haar measure. Similarly, any right translationMathworldPlanetmathPlanetmath on G any right Haar measure.

  • Every continuous surjective homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between compact Hausdorff is measure-preserving relatively to the normalized Haar measure (see this entry (

Title measure-preserving
Canonical name Measurepreserving
Date of creation 2013-03-22 12:19:41
Last modified on 2013-03-22 12:19:41
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 17
Author asteroid (17536)
Entry type Definition
Classification msc 28D05
Classification msc 37A05
Synonym measure preserving
Synonym measure-preserving transformation
Synonym measure-preserving map
Related topic ErgodicTransformation
Defines invertible measure-preserving transformation
Defines endomorphism of a measure space