This is a quick thought that I don’t have time to work on, but thought I’d write down in case it was interesting to anyone.
Recently, @dspivak wrote a blog post on the distributions monad, which is a monad in \mathsf{Poly} defined by
Given a set X, an element of \mathsf{lott}(X) is a finite set of lottery tickets \{1,\ldots,N\}, a function f \colon \{1,\ldots,N\} \to X, and a distribution on \{1,\ldots,N\}. This induces a finitely supported distribution on X, but multiple “lotteries” can induce the same distribution.
This reminded me a lot of the philosophy of quasi-Borel spaces. A quasi-Borel space, roughly speaking, is a set X along with, for every standard Borel space \Omega, a set of “allowable variables” X(\Omega) \subset \{f \colon \Omega \to X\}. More formally, it is a sheaf on the site of standard Borel spaces. It turns out that all uncountable standard Borel spaces are isomorphic, so it suffices just to look at, for instance, \Omega = \mathbb{R}, but I think it’s cleaner to not do it this way.
Anyways, I think you can do a “infinite lottery monad” construction, which sends a quasi-Borel space X to the set of tuples (\Omega, \mu \in \mathcal{P}(\Omega), f \in X(\Omega)), where \mathcal{P}(\Omega) is the set of measures on \Omega.
The reason that we need to consider quasi-Borel spaces is that we want f to be “reasonable” in some sense; i.e. we don’t want some nasty function \mathbb{R} \to \mathbb{R} that needs the axiom of choice to be defined.
I think that if we simply work in \mathsf{Poly} over the quasi-topos of quasi-Borel spaces (David assures me that a quasi-topos has enough structure to do the Poly construction), then the polynomial
works as a good analogue to \mathsf{lott}.
The definition of distributions for quasi-Borel spaces is in fact a quotient of \mathsf{lott}^\infty(X), so I hope that a similar story to David’s blog post could be told.