If the sampling model is a continuous function of the parameter and if either the parameter space or the observation space is compact, then a coherent inference which is a continuous function of the observation must be the posterior of a proper, countably additive prior.
"Coherent and Continuous Inference." Ann. Statist. 11 (1) 114 - 120, March, 1983. https://doi.org/10.1214/aos/1176346062