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?

topology of point-wise convergence.$\endgroup$Stationaryis 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 itinvariant? $\endgroup$