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invariant by a measure-preserving transformation (Definition)

Let $ X$ be a set and $ T:X \longrightarrow X$ a transformation of $ X$.

The notion of invariance by $ T$ we are about to describe is stronger than the usual notion of invariance, and is especially useful in ergodic theory. Thus, in most applications, $ (X, \mathfrak{B}, \mu)$ is a measure space and $ T$ is a measure-preserving transformation. Nevertheless, the definition of invariance and its properties are general and do not require any such assumptions.

$ \,$

Definition - A subset $ A \subseteq X$ is said to be invariant by $ T$, or $ T$-invariant, if $ T^{-1}(A)=A$.

$ \,$

The fundamental property of this concept is the following: if $ A$ is invariant by $ T$, then so is $ X \setminus A$.

Thus, when $ A$ is invariant by $ T$ we obtain by restriction two well-defined transformations

$\displaystyle T\vert _A :A \longrightarrow A$    
$\displaystyle T\vert _{X\setminus A} : X\setminus A \longrightarrow X\setminus A$    

Hence, the existence of an invariant subset allows one to decompose the set $ X$ into two disjoint subsets and study the transformation $ T$ in each of these subsets.

Remark - When $ T$ is a measure-preserving transformation in a measure space $ (X, \mathfrak{B}, \mu)$ one usually restricts the notion of invariance to measurable subsets $ A \in \mathfrak{B}$.



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Keywords:  invariant
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Cross-references: measurable, disjoint, transformations, well-defined, restriction, subset, measure-preserving transformation, measure space, applications, theory, ergodic, transformation
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This is version 1 of invariant by a measure-preserving transformation, born on 2008-05-19.
Object id is 10604, canonical name is InvariantByAMeasurePreservingTransformation.
Accessed 273 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
 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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