TY - JOUR
AU - Krasikova, I.
AU - Pliev, M.
AU - Popov, M.
PY - 2021/04/29
Y2 - 2024/04/20
TI - Measurable Riesz spaces
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 13
IS - 1
SE - Scientific articles
DO - 10.15330/cmp.13.1.81-88
UR - https://scijournals.pnu.edu.ua/index.php/cmp/article/view/4381
SP - 81-88
AB - <p>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.</p><p>(1) The Boolean algebra $\mathcal{U}$ of bands of $E$ is measurable.</p><p>(2) $E_{\rm meas} = E$ and $E$ satisfies the countable chain condition.</p><p>(3) $E$ can be embedded as an order dense subspace of $L_0(\mu)$ for some probability measure $\mu$.</p>
ER -