A subset $S$ of a real vector space is convex if it is closed under finite mixtures: for any $\lambda_1,\ldots,\lambda_n>0$ such that $\lambda_1+\cdots+\lambda_n=1$ and any $x_1,\ldots,x_n\in S$, $\lambda_1x_1+\cdots+\lambda_nx_n\in S$. In a normed space, we can generalize and talk about countable mixtures, defined in the obvious way. I suspect that with still more structure, we can define even more general notions of mixture (using some kind of integration, perhaps). But how does it go? Are there examples of sets closed under countable mixtures but not arbitrary mixtures?
If it helps, what I am most interested in is mixtures of probability measures and mixtures of finitely-additive probability measures.