This article is cited in 1 scientific paper (total in 1 paper)
Maximal quasi-normed extension of quasi-normed lattices
A. G. Kusraevab, B. B. Tasoevc
a North Ossetian State University, 44-46 Vatutin Street, Vladikavkaz, 362025, Russia
b Vladikavkaz Science Center of the RAS, 22 Markus Street, Vladikavkaz, 362027, Russia
c Southern Mathematical Institute — the Affiliate of
Vladikavkaz Science Center of the RAS, 22 Markus street, Vladikavkaz, 362027, Russia
The purpose of this article is to extend the Abramovich's construction of a maximal normed extension of a normed lattice to quasi-Banach setting. It is proved that the maximal quasi-normed extension $X^\varkappa$ of a Dedekind complete quasi-normed lattice $X$ with the weak $\sigma$-Fatou property is a quasi-Banach lattice if and only if $X$ is intervally complete. Moreover, $X^\varkappa$ has the Fatou and the Levi property provided that $X$ is a Dedekind complete quasi-normed space with the Fatou property. The possibility of applying this construction to the definition of a space of weakly integrable functions with respect to a measure taking values from a quasi-Banach lattice is also discussed, since the duality based definition does not work in the quasi-Banach setting.
quasi-Banach lattice, maximal quasi-normed extension, Fatou property, Levi property vector measure, space of weakly integrable functions.
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MSC: 46A16, 46B42, 46E30, 46G10, 47B38, 47G10
A. G. Kusraev, B. B. Tasoev, “Maximal quasi-normed extension of quasi-normed lattices”, Vladikavkaz. Mat. Zh., 19:3 (2017), 41–50
Citation in format AMSBIB
\by A.~G.~Kusraev, B.~B.~Tasoev
\paper Maximal quasi-normed extension of quasi-normed lattices
\jour Vladikavkaz. Mat. Zh.
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A. G. Kusraev, B. B. Tasoev, “Integrirovanie po polozhitelnoi mere so znacheniyami v kvazibanakhovoi reshetke”, Vladikavk. matem. zhurn., 20:1 (2018), 69–85
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