Coherent families of probabilities: properties and extensions

Let B be a Boolean Algebra and S a subset of B. An S-coherent family of probabilities is defined as a family associating to every element of S a probability defined over B, and verifying some coherence conditions. This notion allows to indicate an alternative way to explore conditional probabilities. Indeed, a coherent family of probabilities is naturally attached to every conditional probability. In the paper some conditions are provided, under which an S -coherent family of probabilities can be built up by a finite number of steps and connections between the notions of S- coherent family of probabilities and layers of events are given.