Non-­conglomerability for countably additive measures that are not κ-­additive

Let κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti (1974) and Dubins (1975), subject to several structural assumptions for creating sufficiently many measurable sets, and assuming that κ is not a weakly inaccessible cardinal, we show that each probability that is not κ-­additive has conditional probabilities that fail to be conglomerable in a partition of cardinality no greater than κ. This generalizes our (1984) result, where we established that each finite but not countably additive probability has conditional probabilities that fail to be conglomerable in some countable partition.