Boolean Combinations of Set-Valued Random Variables

Abstract

As we remember, the role of one of the basic building stones in our definition of belief and plausibility functions over infinite sets S was played by a set-valued random variable U, defined on the abstract probability space (Ω,A, P) and taking its values in a measurable space (P(S), S) over the power-set P(S) of all subsets of S. Having at hand two or more such set-valued random variables, an immediate idea arises to define new set-valued random variables, applying boolean set-theoretical operations to the values of the original variables. Namely, let U be a nonempty set of random variables defined on (Cl, A, P) and taking their values in (P(S), S), let U ∈ U. We may define set-valued mappings ∩U, ∪U and S — U setting, for each ω∈Ω, $$\begin{gathered} \left( { \cap ^u } \right)\left( \omega \right) = \cap \{ U\left( \omega \right):u \in u\} , \hfill \\ \left( { \cup ^u } \right)\left( \omega \right) = \cup \{ U\left( \omega \right):u \in u\} , \hfill \\ \left( {S - U} \right)\left( \omega \right) = S - U\left( \omega \right). \hfill \\ \end{gathered} $$ (10.1.1) .