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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, homomorphism, 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 6641 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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