Upper and lower conditional probabilities induced by a multivalued mapping

Given a (finitely additive) full conditional probability space and a conditional measurable space , a multivalued mapping Γ from X to Y induces a class of full conditional probabilities on . A closed form expression for the lower and upper envelopes ⁎ and ⁎ of such class is provided: the envelopes can be expressed through a generalized Bayesian conditioning rule, relying on two linearly ordered classes of (possibly unbounded) inner and outer measures. For every , ⁎ is a normalized totally monotone capacity which is continuous from above if is a countably additive full conditional probability space and is a σ-algebra. Moreover, the full conditional prevision functional M induced by μ on the set of -continuous conditional gambles is shown to give rise through Γ to the lower and upper full conditional prevision functionals ⁎ and ⁎ on the set of -continuous conditional gambles.