I am working on extending the concept of a probability space to the surreal numbers, the main reason which being that I think that having a set whose measure is nonzero while the measure of each element is zero is not intuitive. I would like to formalize the inverse of this, being that every event that could happen contributes something to the probability, within a $\sigma$-algebra. The probability space is $(\Omega, F, P)$ with $\Omega$ and $F$ being sets, and $P: F \rightarrow \mathbf{No}$.
The simplest formulation is $\forall E \in F: P(E)>0 \Rightarrow \exists e \in E: P(\{e\})>0$. However this fails if $\{e\}$ is not in $F$. My next try was to define a set $\bar{E}=\bigcap_{e \in F: e \subseteq E \land P(e)=P(E)}e$, and so the axiom would be
$\forall E \in F: P(E)>0 \Rightarrow \forall \bar{e} \in F, \bar{e} \subseteq \bar{E}, P(\bar{e})>0$.
This may fail in the same way, where a $\sigma$-algebra does not guarantee arbitrary intersections so there may be no subsets of $\bar{E}$ in $F$. It seems much closer. How could I better formalize this without requiring arbitrary unions or intersections?
The condition that the post seems to be trying to articulate is sometimes known as perfect additivity: the requirement that the probability of an arbitrary union of mutually disjoint events is equal to the sum of their separate probabilities. Clearly one has to be careful how to define the sum when it is infinite. So in effect, the idea is to allow for arbitrary unions, and to develop the machinery to handle them.
I have not seen perfect additivity discussed for surreal-valued probabilities. But it has been extensively discussed in the case where probabilities are values in nonarchimedean ordered fields. See, for example, the paper Infinitesimal probabilities by Benci, Horsten, and Wenmackers. Given especially that every ordered field embeds into the surreals (though what exactly that means is complicated: see here and here on MO), looking at that literature would seem a good place to start.