Convergence in Riesz spaces with conditional expectation operators

Abstract

A conditional expectation, \(T\), on a Dedekind complete Riesz space with weak order unit is a positive order continuous projection which maps weak order units to weak order units and has \(R(T)\) a Dedekind complete Riesz subspace of \(E\). The concepts of strong convergence and convergence in probability are extended to this setting as \(T\)-strongly convergence and convergence in \(T\)-conditional probability. Critical to the relating of these types of convergence are the concepts of uniform integrability and norm boundedness, generalized as \(T\)-uniformity and \(T\)-boundedness. Here we show that if a net is \(T\)-uniform and convergent in \(T\)-conditional probability then it is \(T\)-strongly convergent, and if a net is \(T\)-strongly convergent then it is convergent in \(T\)-conditional probability. For sequences we have the equivalence that a sequence is \(T\)-uniform and convergent in \(T\)-conditional probability if and only if it is \(T\)-strongly convergent. These results are applied to Riesz space martingales and are applicable to stochastic processes having random variables with ill-defined or infinite expectation.