Formal definition of ergodicity

Question

I am having a hard time understanding the link between the ergodicity of a random process and the ergodicity of what is called a measure-preserving transformation T as stated in wikipedia here.

Let $(X,\; \Sigma ,\; \mu\,)$ be a probability space, and $T:X \to X$ be a measure-preserving transformation. We say that T is ergodic with respect to $\mu$ (or alternatively that $\mu$ is ergodic with respect to T) if the following equivalent conditions hold:

• for every $E \in \Sigma$ with $T^{-1}(E)=E\,$ either $\mu(E)=0\,$ or $\mu(E)=1\,$;

• for every $E \in \Sigma$ with ${\displaystyle \mu > (T^{-1}(E)\bigtriangleup E)=0}$ we have $\mu(E)=0$ or $\mu(E)=1\,$ (where $\bigtriangleup$ denotes the symmetric difference);

• for every $E \in \Sigma$ with positive measure we have ${\displaystyle \mu \left(\bigcup _{n=1}^{\infty }T^{-n}(E)\right)=1}$;

• for every two sets E and H of positive measure, there exists an n > 0 such that ${\displaystyle \mu ((T^{-n}(E))\cap H)>0}$;

• Every measurable function $f:X\to\mathbb{R}$ with $f\circ T=f$ is almost surely constant.

I actually found a very similar question on SE here:

Definition of ergodicity and ergodic process

But I still can't really picture what does T represent, and what a measure-preserving transformation is in the world of random processes. Has it something to do with a time translation?

Right now, the idea I have in mind of an ergodic random process is a process for which the distribution over the whole signal is somehow similar to the distribution of that very signal at any specific point in time.

An answer with an illustrated example would be very welcome.

Answer 1

This is a bit handwavey and intuitive, but here’s an attempt.

Let’s have $X$ be a deck of cards, $\Sigma = P(X)$ the $\sigma$-algebra of subsets of cards, and $\mu$ the atomic probability measure that assigns each card the same weight.

An example of a measure-preserving transformation $T$ would be shuffling the deck of cards. We can think of this as a bijection of sets $T \colon X \to X$. This is measure-preserving because for $V \subset X$, $\mu(V)$ depends only on the cardinality of $V$. Since $T$ is a bijection, $\mu(T(V)) = \mu(V)$.

When is shuffling cards an ergodic process? The first condition says that $T$ is ergodic if the only sets of cards left invariant by the shuffling are the empty set and the whole deck. The second condition in this instance says the same thing because the empty set is the only measure zero set of cards.

The third condition says, if $E \in \Sigma$ is say a set containing a single card $x$, then $T$ is ergodic if for every other card $y$, there exists $k \ge 0$ such that $T^k(y) = x$; i.e. every other card eventually ends up in the place where $x$ is now. The fourth condition says the same thing in slightly more generality.

The final condition says that if I have a function $f\colon X \to \mathbb R$ that assigns a real number to each card, then if $f(T(x)) = f(x)$ for all $x$, we conclude that $f$ is constant.

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The formal definition only abstracts this picture to the more general setting of an arbitrary probability space.