Abstract: 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 withrespect 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.
Keywords: quasi-Banach lattice, maximal quasi-normed extension, Fatou property, Levi property vector measure, space of weakly integrable functions
For citation: Kusraev A. G., Tasoev B. B. Maximal quasi-normed extension of quasi-normed lattices // Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp. 41-50.
DOI 10.23671/VNC.2017.3.7111
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