Measurable Riesz spaces

Abstract

We study measurable elements of a Riesz space $E$, i.e. elements $e \in E \setminus \{0\}$ for which the Boolean algebra $\mathfrak{F}_e$ of fragments of $e$ is measurable. In particular, we prove that the set $E_{\rm meas}$ of all measurable elements of a Riesz space $E$ with the principal projection property together with zero is a $\sigma$-ideal of $E$. Another result asserts that, for a Riesz space $E$ with the principal projection property the following assertions are equivalent.
 (1) The Boolean algebra $\mathcal{U}$ of bands of $E$ is measurable.
 (2) $E_{\rm meas} = E$ and $E$ satisfies the countable chain condition.
 (3) $E$ can be embedded as an order dense subspace of $L_0(\mu)$ for some probability measure $\mu$.

On linear sections of orthogonally additive operators

“…Orthogonally additive operators on Riesz spaces naturally generalize linear operators, and in recent years a number of results on linear operators were generalized to the orthogonally additive ones by different authors, see e.g. [1,3,5,10,11,12,13,16,19,21] and the bibliography therein. Necessary background for the theory of orthogonally additive operators was prepared by J. M. Mazón and S. Segura de León in [10,11] (1990), and since then a number of mathematicians actively study different problems on orthogonally additive operators on Riesz spaces.…”