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.