Measurable Riesz spaces
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 projecti...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Vasyl Stefanyk Precarpathian National University
2021-04-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Subjects: | |
| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/4381 |
| Summary: | 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 $\mu$. |
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| ISSN: | 2075-9827 2313-0210 |