# Stationary, ergodic measures from the structuralist point of view

Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random variables. I am working on a model from mathematical physics involving such measures, and I would like to express them in the language of Dmitri Pavlov's localizable measurable spaces.

Let $Q = \mathbb Z^d$, and let $X$ be a finite set. Consider the space $S = X^Q$, equipped with the product topology. Since $X$ is finite, $X^Q$ is compact.

Elements of $S$ are spin fields, and there is the natural isomorphism from $S$ to $S' = C(Q,X)$. There is probably a nice name for the topology on $S'$ induced by the product topology $S$, but I don't know what it is.

Question 1: The product topology on $S = X^Q$ corresponds to what topology on $S' = C(Q, X)$?

Let $G \cong \mathbb Z^d$ be the translation group of $Q$. This naturally acts on $S$. i.e., if $\tau_v \in G$, then $\tau_v(s)(q) := s(v + q).$

Let $\mathcal B(S)$ denote the Borel $\sigma$-algebra of $S$. Following the perspective of Dmitri Pavlov, I would like to turn $S$ into a localizable measurable space so that I can consider families of measures on $S$.

To do this, I need a natural $\sigma$-ideal $\mathcal N(S)$ which is closed under the action of $G$, and for which the quotient $\mathcal B(S) / \mathcal N(S)$ is a complete Boolean lattice.

Question 2: Does there exist a natural $\sigma$-ideal so that $(S, \mathcal B(S), \mathcal N(S))$ is a localizable measurable space?

Maybe this is too general; the classical existence of many measures on $S$ suggets that there are lots of such $\sigma$-ideals! (take any complete measure, and let $\mathcal N$ be its collection of null sets)

However, I want to build ergodicity into the definition too. We say that $\mathcal N$ is an ergodic $\sigma$-ideal when $$A \triangle \tau^{-1} A \in \mathcal N \mathrm{~for~all~} \tau \in G \mathrm{~implies~} A \in \mathcal N \mathrm{~or~} S-A \in \mathcal N.$$ That is, if a measurable set $A$ is effectively translation-invariant, then it must have either zero or full measure.

Supposing that such an ergodic $\sigma$-ideal exists, let $M(S)$ denote the space of real-valued ergodic measures on $(S, \mathcal B(S), \mathcal N(S))$. Measures push-forward, so the group $G$ naturally acts on $M(S)$. We say that a measure is stationary if it is invariant under translations. i.e., $\mu = \tau_* \mu$ for all $\tau \in G$.

Question 3: What is the structure of the space of stationary, ergodic measures?

• Is $C(Q,X)$ the space of all continuous functions? Since $Q$ is discrete it is not only isomorphic but rather identical to the space $X^Q$ of all functions from $Q$ to $X$. The product topology is the topology of point-wise convergence. – Jochen Wengenroth Oct 11 '12 at 7:00
• Stationary is an infelicitous name, as there are specific dynamical systems (e.g., shifts of finite type) to which stationary can be applied (in the sense that the transition matrix---or something like it---is the same). Why not just call it invariant? – David Handelman Apr 17 '16 at 21:13
• When $d=1$, every metrizable Choquet simplex can arise. (Q3) – David Handelman Apr 17 '16 at 21:16

I'm not familiar enough with the notion of localisable measurable space in Pavlov's sense to say anything too authoritative, but I can make the following comments which apply to at least the case $d=1$ (as a dynamicist working mostly with group actions generated by a single continuous map $f\colon X\to X$ this is the natural setting for me):
1. An ergodic measure is uniquely defined by its collection of null sets, in the following sense: if $\mu$ is an ergodic stationary measure and $\nu\ll \mu$ is also ergodic and stationary, then $\nu=\mu$. This is just because the Radon-Nikodym derivative $d\nu/d\mu$ is an invariant function and hence a.e.-constant.
2. As discussed in this other question, there are many examples of dynamical systems (including in particular full shift spaces, which is the dynamicist's way of talking about the set of spin fields when $d=1$) for which the space of stationary ergodic measures is the set of extreme points of a Poulsen simplex. In particular it is path-connected and dense in its convex closure.