This paper studies belief functions, set functions which are normalized and monotone of order $\infty$. The concepts of continuity and condensability are defined for belief functions, and it is shown how to extend continuous or condensable belief functions from an algebra of subsets to the corresponding power set. The main tool used in this extension is the theorem that every belief function can be represented by an allocation of probability--i.e., by a $\cap$ -homomorphism into a positive and completely additive probability algebra. This representation can be deduced either from an integral representation due to Choquet or from more elementary work by Revuz and Honeycutt.
"Allocations of Probability." Ann. Probab. 7 (5) 827 - 839, October, 1979. https://doi.org/10.1214/aop/1176994941