Definition:Ergodic Invariant Measure

Definition
Let $\struct {X, \BB}$ be a measurable space.

Let $T: X \to X$ be a measurable mapping.

Let $\mu$ be a $T$-invariant probability measure on $\struct {X, \BB}$.

Then $\mu$ is said to be ergodic :
 * $T$ is an ergodic transformation on $\struct {X, \BB, \mu}$

Also known as
More completely, it is called ergodic $T$-invariant measure.