# Intuitive meaning of Giry monad's $\sigma$-algebra

The Giry monad $$G : \textbf{Meas} \to \textbf{Meas}$$ maps a measurable space $$(X, \mathcal{F})$$ to its set of probability measures. The $$\sigma$$-algebra of $$G(X, \mathcal{F})$$ is the smallest algebra such that the evaluation map $$p \mapsto p(E)$$ is measurable for all $$E \in \mathcal{F}$$. I am aware that this condition is needed to define the integral in $$\mu_X : \mathcal{G}^2X \to \mathcal{G}X$$, but what is the intuitive meaning of the $$\sigma$$-algebra? Is there an intuitive example of a subset $$A \subseteq \mathcal{G}(X, \mathcal{F})$$ that is not in the $$\sigma$$-algebra?

Here are two kinds of sets which typically aren’t in the Giry $$\sigma$$-algebra $$G(X)$$:
• $$\{ p \mid p(E) \in A \}$$, for some measurable $$E \subseteq X$$ and non-measurable $$A \subseteq [0,1]$$. (For specific spaces $$X$$, it’s not hard to check this is non-measurable; I guess that should be true in fairly wide generality, but I don’t quickly see how to prove that.)
• $$\{ p \mid p(E_i) \in A_i,\, \text{for all}\ i \in I \}$$, for some arbitrary uncountable family of measurable sets $$E_i \subseteq X$$, $$A_i \subseteq [0,1]$$. For instance, take $$X$$ to be an uncountable set with the $$\sigma$$-algebra of just its countable and co-countable subsets. Then it’s not hard to check that for every measurable set $$S$$ of $$G(X)$$, the set $$\{ p(\{x\}) \mid p \in S \}$$ is an interval $$[0,a]$$ for all but countably many $$x \in X$$.
My overall intuition is that very crudely, the Giry $$\sigma$$-algebra is a bit like the (infinite-indexed) product topology: on the one hand there’s a “componentwise” constraint, as exemplified by the former counterexample above; on the other hand theres an “across-components” constraint, as seen with the latter.
• Note that a natural example falling into the second category are singletons $\{p\}$ which aren't in the $\sigma$-algebra if $\mathcal{F}$ isn't countably generated. Jun 10, 2023 at 13:59