# Measurable Riesz spaces

## Authors

• I. Krasikova Zaporizhzhya National University, 66 Zhukovskoho str., 69002, Zaporizhzhya, Ukraine https://orcid.org/0000-0002-7559-3758
• M. Pliev Southern Mathematical Institute of the Russian Academy of Sciences, 22 Markusa str., North-Ossetian State University, 44-46 Vatutina str., 362025 Vladikavkaz, Russia https://orcid.org/0000-0001-8835-8805
• M. Popov Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine, Pomeranian University in Słupsk, 76-200, Słupsk, Poland https://orcid.org/0000-0002-3165-5822

## Keywords:

vector lattice, Riesz space, Boolean algebra of bands
Published online: 2021-04-29

### 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 $\mu$.

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How to Cite
(1)
Krasikova, I.; Pliev, M.; Popov, M. Measurable Riesz Spaces. Carpathian Math. Publ. 2021, 13, 81-88.