On coherent conditional probabilities and disintegrations

Existence of coherent extensions of coherent conditional probabilities is one of the major merits of de Finetti's theory of probability. However, coherent extensions which meet some special property, like sigma-additivity or disintegrability, can fail to exist. An example is given where a coherent and sigma-additive conditional probability cannot be extended preserving both sigma-additivity and coherence. Motivated by such example, conditions are provided in order that a coherent and sigma-additive conditional probability admits a coherent and sigma-additive extension. Moreover, conditions are given for the existence of disintegrations, possibly sigma-additive, of a probability along a partition.