Nonconglomerability 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 probabililties that fail to be conglomerable in some countable partition. (ii) P(B | B) = 1 Moreover, following de Finetti (1974) and Dubins (1975), in order to regulate conditional probability given a non-­‐empty event of unconditional or conditional probability 0, we require the following. As is usual, we identify the unconditional probability function P(⋅) with P(⋅ | Ω) and refer to P(⋅) as a probability function. This account of conditional probability is not the usual theory from contemporary Mathematical Probability. It differs from the received theory of Kolmogorovian regular conditional distributions in four ways: 1. The theory of regular conditional distributions requires that probabilities and conditional probabilities are countably additive. The de Finetti/Dubins theory requires only that probability and conditional probability is finitely additive. In this paper, we bypass most of this difference by exploring de Finetti/Dubins conditional probabilities associated with countably additive unconditional probabilities. Specifically, we do not require that conditional probabilities are countably additive. 2. When P(B) = 0 and B is not empty, a regular conditional probability given B is relative to a sub-­‐σ-­‐field A ⊆ B, where B ∈ A. In the Finetti/ Dubins theory of conditional probability, P(⋅ | B), depends solely on the event B and not on any