We discuss here several types of convergence of conditional expectations for unbounded closed convex random sets of the form where is a decreasing sequence of sub-σ-algebras and is a sequence of closed convex random sets in a separable Banach space.
Let (X, F, μ) be a finite measure space. Let T: X → X be a measure preserving transformation and let Anf denote the average of Tkf, k = 0, ..., n. Given a real positive function v on X, we prove that {Anf} converges in the a.e. sense for every f in L1(v dμ) if and only if infi ≥ 0 v(Tix) > 0 a.e., and the same condition is equivalent to the finiteness of a related ergodic power function Prf for every f in L1(v dμ). We apply this result to characterize, being T null-preserving, the finite...
In some recent papers, results of uniform additivity have been obtained for convergent sequences of measures with values in -groups. Here a survey of these results and some of their applications are presented, together with a convergence theorem involving Lebesgue decompositions.
In this paper we prove two convergence theorems for set-valued conditional expectations. The first is a set-valued generalization of Levy’s martingale convergence theorem, while the second involves a nonmonotone sequence of sub -fields.
We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence of functions from a measure space to a Banach space. In one result the equi-integrability of ’s is involved and we assume almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of to is assumed.
We give a definition of uniform PU-integrability for a sequence of -measurable real functions defined on an abstract metric space and prove that it is not equivalent to the uniform -integrability.
We obtain for measures on MV-algebras the classical theorem of Dieudonné related to convergent sequences of regular maps.
Assuming the continuum hypothesis, we show that (i) there is a compact convex subset L of , and a probability Radon measure on L which has no separable support; (ii) there is a Corson compact space K, and a convex weak*-compact set M of Radon probability measures on K which has no -points.