What is the Eilenberg-Moore category for the non-finitary probability distribution monad is, that is, the monad $D \colon \mathbf{Set} \to \mathbf{Set}$ defined by


$$
DX = \left\{ p \in [0,1]^X \ \Big|\ \left|p^{-1}(0, 1]\right| \le \aleph\_0,\ \sum\_{x \in X} p(x) = 1 \right\}?
$$


I can only find information on the finitary version, e.g. [here](http://www.cs.ru.nl/B.Jacobs/PAPERS/convex-algebra-effect.pdf).



Algebras for probability monad
The algebras of this monad and closely related ones have been introduced by Pumplün and Röhrl. For example the introduction of [a paper](http://link.springer.com/article/10.1007%2FBF00122684) by Börger and Kemper provides a good summary,



> 
> Pumplün and Röhrl [6] introduced the notion of totally convex space. These spaces are the Eilenberg-Moore-algebras for the monad induced by the unit-ball functor $\bigcirc:\mathbf{Ban}\_1\to \mathbf{Set}$, where $\mathbf{Ban}\_1$ is the category of Banach spaces and linear operations of norm $\leq 1$. They can be described as sets with formal infinitary linear combination maps $X^\mathbb{N}\to X,\: (x\_i)\_{i\in\mathbb{N}} \mapsto \sum\_{i\in\mathbb{N}}\alpha\_i x\_i$, where $\alpha\_i\in K$, $\sum\_{i\in\mathbb{N}}|\alpha\_i|\leq 1$, $K\in\{\mathbb{R},\mathbb{C}\}$, subject to two axioms.
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> Pumplün and Röhrl also considered the categories obtained by restricting the operations to positive coefficients (maybe even with $\sum\_{i\in\mathbb{N}}\alpha\_i=1$) and/or finite arity ([4,5,8]). [..]
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The relevant reference here is [4], which points to Pumplün's paper "Banach spaces and superconvex modules". The latter seem to be the algebras of your monad, as are the "superconvex spaces" of the paper linked to above. (Although there is a cryptic remark on whether the empty set counts as superconvex or not.) I haven't been able to track down reference [4], but a web search for "superconvex module" and "superconvex space" reveals some other papers.


Finally, it may be worth emphasizing that your monad probably arises from a very natural adjunction similar to the one formed by the unit ball functor $\bigcirc:\mathbf{Ban}\_1\to\mathbf{Set}$ and its left adjoint. I haven't checked the details, but I bet that it is the monad induced by the forgetful functor from the category of norm-complete [base norm spaces](http://link.springer.com/chapter/10.1007/3-540-06725-6_4) with base-preserving positive linear maps as morphisms to $\mathbf{Set}$. Maybe you already know?


