Definition:Ergodic Measure-Preserving Transformation

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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