Generalization of the theorems of Barndorff-Nielsen and Balakrishnan-Stepanov to Riesz spaces

Abstract

In a Dedekind complete Riesz space, $E$, we show that if $(P_n)$ is a sequence of band projections in $E$ then $$\underset{n\to \infty }{\mathrm{limsup}}{P}_n-\underset{n\to \infty }{\mathrm{liminf}}{P}_n=\underset{n\to \infty }{\mathrm{limsup}}{P}_n(I-P_{n+1}).$$ This identity is used to obtain conditional extensions in a Dedekind complete Riesz spaces with weak order unit and conditional expectation operator of the Barndorff-Nielsen and Balakrishnan-Stepanov generalizations of the First Borel-Cantelli Theorem.