Upper and lower conditional probabilities induced by a multivalued mapping

Given a (finitely additive) full conditional probability space (X, F x F-0, mu) and a conditional measurable space (Y, G x G(0)), a multivalued mapping Gamma from X to Y induces a class of full conditional probabilities on (Y, G x G(0)). A closed form expression for the lower and upper envelopes mu(*) and mu* 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 is an element of G(0), mu(*)(.vertical bar B) is a normalized totally monotone capacity which is continuous from above if (X, F x F-0, mu) is a countably additive full conditional probability space and is a a sigma-algebra. Moreover, the full conditional prevision functional M induced by mu on the set of F-continuous conditional gambles is shown to give rise through Gamma to the lower and upper full conditional prevision functionals M-* and M* on the set of G-continuous conditional gambles. For every B E G(0), M-* (.vertical bar B) is a totally monotone functional having a Choquet integral expression involving mu(*). Finally, by considering another conditional measurable space (Z, H x H-0) and a multivalued mapping from Y to Z, it is shown that the conditional measures mu(**), mu** and functionals M-**, M** induced by mu(*), preserve the same properties of mu(*), mu* and M-*,M- M*. (C) 2017 Elsevier Inc. All rights reserved.