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 $\struct {X, \BB}$.

This article, or a section of it, needs explaining. In particular: Does it actually have to be a probability measure? Yes, one considers only probability invariant measures in ergodic theory, because it is the study of statistical behaviors in dynamical systems. In more advanced sub-branches, one also considers $\sigma$-finite measures (infinite ergodic theory) or non-invariant ergodic measures. But these should be ignored at basic level. We don't limit ourselves to the basic level on $\mathsf{Pr} \infty \mathsf{fWiki}$. If there is a "basic" option that is used, we state this separately. The definitions given here are classical and standard in ergodic theory and the formulations should be sufficiently general for non-exparts. For infinite ergodic theory or something else, it is better to create a separate category because that is a completely different world (jungle) and too immature. Almost nothing remains common with the classical theory. Sorry, may word choice basic was confusing. Please consider the use of discussion page. What's a non-expart? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. 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 {{Explain}} from the code.

Then $\mu$ is said to be ergodic if and only if:

$T$ is an ergodic transformation on $\struct {X, \BB, \mu}$

Also known as

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

Sources

• 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $2.3$: Ergodicity