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

In a Dedekind complete Riesz space, E, we show that if $$(P_n)$$ is a sequence of band projections in E then $$\begin{aligned} \limsup \limits _{n\rightarrow \infty } P_n - \liminf \limits _{n\rightarrow \infty } P_n = \limsup \limits _{n\rightarrow \infty } P_n(I-P_{n+1}). \end{aligned}$$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.