Let $O$ be a set of objects and $S$ a set of states. Every object is in a state, which is described by a function $\sigma\colon O \to S$. Such a function is called a configuration.
In this setting I want to study probabilities of configurations, so I consider measures $\mu \colon [O\to S] \to {\Bbb R}$ on the set $[O\to S]$ of functions from $O$ to $S$. (Or, rather, on a suitable $\Sigma$-algebra on $[O\to S]$) I want to express the concept that objects are anonymous or homogeneous in the sense that the specific identity of an object is irrelevant. More formally, for every bijection $f\colon O \to O$ on the objects and function $\sigma\colon O\to S$ I expect $\mu(\sigma) = \mu (\sigma \circ f)$ to hold (or, rather, on the $Ssigma$-algebra).
I assume that this requirement eventually results in $\mu$ being a product measure of a measure $\nu\colon S \to {\Bbb R}$ in the sense $\mu = \prod_{o\in O}\nu $ while identifying $[O\to S]$ with $\prod_{o\in O}S$.
Essentially this boils down to the intuition: If the probability of a configuration is independent on which particular object is in which state, then the probability of a configuration depends only on the distribution of the states which are present in a configuration.
I need this for solving an applied physical problem - not being an expert in measure I look for pointers for theorems of that kind. My problem in particular is: How would I handle the non-finite case where I really need $\Sigma$ algebras?
Question: Which results and proofs are known in this area?