What's the difference between an ergodic measure, ergodic sequence or ergodic mapping?

Question

So I work a lot with ergodic theorems but it is a bit confusing. We have a probability space $(\Omega,\mathcal{F},P\}$ and a measure preserving mapping $T:\Omega\rightarrow\Omega$. We say that $T$ is ergodic (or $P$-ergodic) if any event in the $T$-invariant $\sigma$-algebra has probability $1$ or $0$. This is not entirely clear to me. As example consider a simple random walk in a random environment, with $\{\omega_i\}_{i\in\mathbb{Z}}$ and $\omega_i$ the probability to jump to the right at position $i$. Now how do I show that this sequence is ergodic? There is no mapping $T$ involved.

Answer 1

The notions of ergodic measure and ergodic mapping are more or less interchangeable. If you already have a probability space $(\Omega, \mathcal{F}, \mathbb{P}),$ you would say that a measure-preserving transformation $T: X \rightarrow X$ is ergodic if $\mathbb{P}(A) \in \{0, 1\}$ for all $A \in \mathcal{F}$ for which $T^{-1}(A) = A$ (this is one of many equivalent definitions). You could also say in this scenario that $\mathbb{P}$ is ergodic (with respect to $T$). However, there are situations in which you are only given a mapping $T$ and are asked if there exists an ergodic measure with respect to $T$ (a very well known theorem shows that if $X$ is a complete metric space and $T$ is continuous, then indeed there exists an ergodic measure). So it really depends on the context, really. As for ergodic sequences, I believe that the wikipedia page on ergodic sequences is a good reference. I hope this helps. :)