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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 $ T$ 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 $ T$ is bijective, measure-preserving, and its inverse $ T^{-1}$ is also measure-preserving, then $ T$ 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.



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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 MSC28D05 (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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