measure-preserving

1 Definition

Definition - Let $(X_{1},\mathfrak{B}_{1},\mu_{1})$ and $(X_{2},\mathfrak{B}_{2},\mu_{2})$ be measure spaces, and $T:X_{1}\to X_{2}$ be a measurable transformation. The transformation $T$ is said to be measure-preserving if for all $A\in\mathfrak{B}_{2}$ we have that

\mu_{1}(T^{-1}(A))=\mu_{2}(A),

where $T^{-1}(A)$ is, as usual, the set of points $x\in X_{1}$ such that $T(x)\in A$ .

Additional Notation:

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If $T$ is bijective, measure-preserving, and its inverse $T^{-1}$ is also measure-preserving, then $T$ is said to be an measure-preserving transformation.

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

Remarks:

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The fact that a map $T:X_{1}\longrightarrow X_{2}$ is measure-preserving depends heavily on the sigma-algebras $\mathfrak{B}_{i}$ and measures $\mu_{i}$ involved. If other measures or sigma-algebras are also in consideration, one should make clear to which measure space is $T:X_{1}\longrightarrow X_{2}$ measure-preserving.

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

2 Properties

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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.

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Let $(X_{1},\mathfrak{B}_{1},\mu_{1})$ and $(X_{2},\mathfrak{B}_{2},\mu_{2})$ be measure spaces and $(X_{1},\overline{\mathfrak{B}_{1}},\overline{\mu_{1}})$ and $(X_{2},\overline{\mathfrak{B}_{2}},\overline{\mu_{2}})$ their completions. If $T:(X_{1},\mathfrak{B}_{1},\mu_{1})\longrightarrow(X_{2},\mathfrak{B}_{2},\mu_{% 2})$ is measure-preserving, then so is $T:(X_{1},\overline{\mathfrak{B}_{1}},\overline{\mu_{1}})\longrightarrow(X_{2},% \overline{\mathfrak{B}_{2}},\overline{\mu_{2}})$ .

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Let $(X_{1},\mathfrak{B}_{1},\mu_{1})$ and $(X_{2},\mathfrak{B}_{2},\mu_{2})$ be measure spaces and $T_{1}:X_{1}\longrightarrow X_{1}$ , $T_{2}:X_{2}\longrightarrow X_{2}$ be measure-preserving maps. Then, the product map $T_{1}\times T_{2}:X_{1}\times X_{2}\longrightarrow X_{1}\times X_{2}$ , defined by

$\displaystyle T_{1}\times T_{2}\;(x_{1},x_{2}):=(T_{1}(x_{1}),T_{2}(x_{2}))$

is a measure-preserving transformation of $(T_{1}\times T_{2},\mathfrak{B}_{1}\times\mathfrak{B}_{1},\mu_{1}\times\mu_{2})$ .

3 Examples

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The identity map of a measure space $(X,\mathfrak{B},\mu)$ is always measure-preserving.

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Let $G$ be a locally compact group (http://planetmath.org/TopologicalGroup). For each $a\in G$ , the transformation $T(g):=ag$ is measure-preserving relatively to any left Haar measure. Similarly, any right translation on $G$ any right Haar measure.

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Every continuous surjective homomorphism between compact Hausdorff is measure-preserving relatively to the normalized Haar measure (see this entry (http://planetmath.org/ContinuousEpimorphismOfCompactGroupsPreservesHaarMeasure)).

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