Embeddability conditions for some spaces (a categorical approach) (Q1060384)

The author proves that some basic properties of FH-spaces [Th. 1,2 from the paper of \textit{A. Wilansky} and \textit{K. Zeller}, Mich. Math. J. 6, 349-357 (1959; Zbl 0103.328)] are valid also in case of so called Zeller's spaces. After that, he applies these resuls of the weighted \(L_ p\)-spaces and proves the following results: Theorem 6. Let \(\{r_ k\}\) be a sequence of measurable positive functions, let \(\{p_ k\}\) be a sequence of positive functions and Y- locally bounded Zeller's space (e.g. \(Y=L_ s)\). Then \(\cap^{\infty}_{k=1}r_ kL_{p_ k}\subseteq Y\) takes place iff \(\cap^{m}_{k=1}r_ kL_{p_ k}\subseteq Y\) for some m. Theorem 7. Let \(\{r_ k\}\) be a sequence of measurable and positive functions which is non increasing and let \(0<s\leq \infty\). Then \(\cap^{\infty}_{m=1}r_ mL_{\infty}\subseteq L_ s\) takes place iff \(r_ mL_{\infty}\subseteq L_ s\) for some m. Theorem 8. Let \(\{r_ m\}\) be an arbitrary sequence of positive measurable functions, \(0<p_ m\leq \infty\) (m\(\in {\mathbb{N}})\). Let X be some Zeller's space. Then the inclusion \(X\subseteq \cup^{\infty}_{m=1}r_ mL_{p_ m}\) takes place iff \(X\subseteq r_ mL_{p_ m}\) for some m.