Definition:Ergodic Measure-Preserving Transformation

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\struct {X, \BB, \mu}$ be a probability space.

Let $T: X \to X$ be a measure-preserving transformation.

Definition 1

$T$ is said to be ergodic if and only if:

for all $A \in \BB$:
$T^{-1} \sqbrk A = A \implies \map \mu A \in \set {0, 1}$


Definition 2

$T$ is said to be ergodic if and only if:

for all $A \in \BB$:
$\map \mu {T^{-1} \sqbrk A \symdif A} = 0 \implies \map \mu A \in \set {0, 1}$

where $\symdif$ denotes the symmetric difference.


Definition 3

$T$ is said to be ergodic if and only if:

for all $A \in \BB$:
$\ds \map \mu A > 0 \implies \map \mu {\bigcup_{n \mathop = 1}^\infty T^{-n} \sqbrk A} = 1$


Definition 4

$T$ is said to be ergodic if and only if:

for all $A, B \in \BB$:
$\map \mu A \map \mu B > 0 \implies \exists n \ge 1 : \, \map \mu {T^{-n} \sqbrk A \cap B} > 0$


Definition 5

$T$ is said to be ergodic if and only if:

for all measurable $f: X \to \C$:
$f \circ T = f$ holds $\mu$-almost everywhere
$\implies \exists c \in \C:\, f = c$ holds $\mu$-almost everywhere


Also see

  • Results about ergodic measure-preserving transformations can be found here.


Sources


This page may be the result of a refactoring operation.
As such, the following source works, along with any process flow, will need to be reviewed.
When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.
In particular: Source citations need to go on the pages holding the actual definitions they provide
If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{SourceReview}} from the code.


Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Ergodic_Measure-Preserving_Transformation&oldid=714326"