Maximal Quasi-Normed Extension of Quasi-Normed Lattices
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<sup>ϰ</sup> of a Dedekind complete quasi-normed lattice X with the weak σ-Fatou property is a quasi-Banach lattice if and only if X is intervally complete. Moreover, X<sup>ϰ</sup> 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.