Upper and lower conditional probabilities induced by a multivalued mapping

Given a (finitely additive) full conditional probability space (X,F×F0,μ) and a conditional measurable space (Y,G×G0), a multivalued mapping Γ from X to Y induces a class of full conditional probabilities on (Y,G×G0). A closed form expression for the lower and upper envelopes μ⁎ and μ⁎ 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 B∈G0, μ⁎(⋅|B) is a normalized totally monotone capacity which is continuous from above if (X,F×F0,μ) is a countably additive full conditional probability space and F is a σ-algebra. Moreover, the full conditional prevision functional M induced by μ on the set of F-continuous conditional gambles is shown to give rise through Γ to the lower and upper full conditional prevision functionals M⁎ and M⁎ on the set of G-continuous conditional gambles. For every B∈G0, M⁎(⋅|B) is a totally monotone functional having a Choquet integral expression involving μ⁎. Finally, by considering another conditional measurable space (Z,H×H0) and a multivalued mapping from Y to Z, it is shown that the conditional measures μ⁎⁎, μ⁎⁎ and functionals M⁎⁎, M⁎⁎ induced by μ⁎ preserve the same properties of μ⁎,μ⁎ and M⁎, M⁎.