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measure-preserving

(Definition)

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 is said to be measure-preserving if for all $A \in \mathfrak{B}_2$ we have that

$\displaystyle \mu_1(T^{-1}(A)) = \mu_2(A),$

where $T^{-1}(A)$ is, as usual, the set of points $x\in X_1$ such that $T(x)\in A$ .

Additional Notation:

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

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

Remarks:

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.

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

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 $(X_1, \mathfrak{B}_1, \mu_1)$ 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 $T:(X_1, \overline{\mathfrak{B}_1}, \overline{\mu_1}) \longrightarrow (X_2, \overline{\mathfrak{B}_2}, \overline{\mu_2})$ .

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 $(T_1 \times T_2, \mathfrak{B}_1 \times \mathfrak{B}_1, \mu_1 \times \mu_2)$ .

Examples

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

Let be a locally compact group. For each $a \in G$ , the transformation is measure-preserving relatively to any left Haar measure. Similarly, any right translation on preserves any right Haar measure.

Every continuous surjective homomorphism between compact Hausdorff groups is measure-preserving relatively to the normalized Haar measure (see this entry).

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"measure-preserving" is owned by asteroid. [ full author list (3) | owner history (3) ]

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See Also: ergodic

Other names:

measure preserving, measure-preserving transformation, measure-preserving map

Also defines:

invertible measure-preserving transformation, endomorphism of a measure space

Attachments:: operator induced by a measure preserving map (Definition) by asteroid

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Cross-references: Haar measure, Hausdorff, compact, surjective, continuous, right Haar measure, translation, right, left Haar measure, locally compact, identity map, product map, completions, codomains, domains, composition, objects, category, morphisms, clear, measures, sigma-algebras, map, inverse, bijective, points, transformation, measurable, measure spaces
There are 12 references to this entry.

This is version 14 of measure-preserving, born on 2002-02-14, modified 2008-05-18.
Object id is 1950, canonical name is MeasurePreserving.
Accessed 6488 times total.

Classification:

AMS MSC:	28D05 (Measure and integration :: Measure-theoretic ergodic theory :: Measure-preserving transformations)
	37A05 (Dynamical systems and ergodic theory :: Ergodic theory :: Measure-preserving transformations)

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