Finite additivity, atomlessness and countable additivity So, I'm trying to get my head around when you can have finitely but not countably additive probabilities. 
The standard example of a finitely additive but not countably additive space is the following strange distribution over the natural numbers.  All finite sets get measure 0, but the whole space gets measure 1. This is finitely additive but not countably so, since a finite union of finite sets is finite, but a countable union needn't be so.
So this got me thinking that if you had an atomless space, examples of this form would be harder to come by. Does atomlessness plus finite additivity guarantee countable additivity? If not, what is missing?
I know that Villegas (1964) shows that for a comparative probability structure to be countably additive, the important properties of the structure are atomlessness and a certain kind of continuity. But I don't know how relevant that point is to the current question.
 A: Here’s an example.
Define an equivalence relation $\sim$ on $\wp(\Bbb N)$ by $A\sim B$ iff $A\,\triangle\, B$ is finite, where $\triangle$ is symmetric difference, and let $\mathscr{B}=\wp(\Bbb N)/\sim$. For $A\subseteq\Bbb N$ denote by $[A]$ the $\sim$-equivalence class of $A$. Let $\mathscr{U}$ be a free ultrafilter on $\Bbb N$. Note that for any $A\subseteq\Bbb N$, $A\in\mathscr{U}$ iff $[A]\subseteq\mathscr{U}$. Now define a $\{0,1\}$-valued measure $\mu$ on $\mathscr{B}$ by $\mu\big([A]\big)=1$ iff $A\in\mathscr{U}$. Then $\mathscr{B}$ is atomless, and $\mu$ is finitely additive. However, $\mu$ is not countably additive, since it is possible to partition $\Bbb N$ into countably infinitely many infinite sets, none of which is in $\mathscr{U}$.
Added: Michael Greinecker has pointed out that I’m using a notion of atomless that may be different from the one intended by Seamus. Here’s another example that may be preferable.
Let $d:\wp(\Bbb N)\to[0,1]$ be asymptotic density, and let $\mathscr{U}$ be a free ultrafilter on $\Bbb N$. For $A\subseteq\Bbb N$ let $$\mu(A)=\mathscr{U}\text{-}\lim_n\frac{|A\cap\{1,\dots,n\}|}n\;.$$ (For basic information on $\mathscr{U}$-limits see this answer by Martin Sleziak.) Then $\mu$ is a finitely additive non-atomic probability measure on $\wp(\Bbb N)$ such that $\mu(A)=d(A)$ whenever $A$ has an asymptotic density.
