This article addresses the problem of existence of a countably additive probability measure in the sense of Kolmogorov that is consistent with a probability assignment to a family of sets which is coherent in the sense of De Finetti.

A dynamic programming principle is derived for a discrete time Markov control process taking values in a finite dimensional space, with ergodic cost and partial observations. This uses the embedding of the process into another for which an accessible atom exists and hence a coupling argument can...