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.
@article{1176346062,
author = {Lane, David A. and Sudderth, William D.},
title = {Coherent and Continuous Inference},
journal = {Ann. Statist.},
volume = {11},
number = {1},
year = {1983},
pages = { 114-120},
language = {en},
url = {http://dml.mathdoc.fr/item/1176346062}
}
Lane, David A.; Sudderth, William D. Coherent and Continuous Inference. Ann. Statist., Tome 11 (1983) no. 1, pp. 114-120. http://gdmltest.u-ga.fr/item/1176346062/