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 $\mu$.