Convergence theorems for set-valued conditional expectations. (English) Zbl 0788.60021

Papageorgiou, Nikolaos S.

Commentat. Math. Univ. Carol. 34, No. 1, 97-104 (1993).

Let \((\Omega,\Sigma,\mu)\) be a probability space, \(X\) a Banach space, and \(F\) a random set in \(X\), i.e. a measurable multifunction from \(\Omega\) to \(X\). For a sub-\(\sigma\)-field \(\Sigma_ 0\) of \(\Sigma\), \(E^{\Sigma_ 0}F\) denotes the conditional expectation of \(F\) with respect to \(\Sigma_ 0\), in the sense of F. Hiai and H. Umegaki [J. Multivariate Anal. 7, 149-182 (1977; Zbl 0368.60006)]. The author proves two convergence theorems for such set-valued conditional expectations. Let \((\Sigma_ n)\) be a sequence of sub-\(\sigma\)-fields of \(\Sigma\). The first theorem says that if \((\Sigma_ n)\) is increasing, then \(E^{\Sigma_ n}F\) converges to \(E^{\Sigma_ 0}F\), where \(\Sigma_ 0\) is generated by \(\bigcup\{\Sigma_ n: n\geq 1\}\). The second theorem deals with nonmonotone \((\Sigma_ n)\) and convergent sequence \((F_ n)\) of random sets. If \(F_ n\to F\) and \(\Sigma_ n\to \Sigma_ 0\) in \(L^ 1(X)\), then the set of integrable selectors of \(E^{\Sigma_ n}F_ n\) converges to the set of integrable selectors of \(E^{\Sigma_ 0}F\). All convergences of sets are in the Kuratowski-Mosco sense.

Reviewer: A.Nowak (Katowice)

MSC:

60D05	Geometric probability and stochastic geometry
60F99	Limit theorems in probability theory
28B20	Set-valued set functions and measures; integration of set-valued functions; measurable selections

Keywords:

Kuratowski-Mosco convergence; random set; convergence theorems; set- valued conditional expectations; integrable selectors

Citations:

Zbl 0368.60006