On the lattice theory of function semi-norms (Q788958)

\textit{W. A. J. Luxemburg} and \textit{A. C. Zaanen} had investigated various properties regarding function semi-norm in the early 60's [Indag. Math. 25, 135-153 (1963; Zbl 0117.080)] but they also left quite a few questions untouched. In this paper the authors looked at the lattice structures of function semi-norms \(\rho\) on the set \({\mathcal M}^+\) of extended real-valued nonnegative measurable functions an a fixed sigma- finite masure space (X,S,\(\mu)\). For a family of semi-norms \(\{\rho_ j:\) \(j\in J\}\) its supremum is defined as \((Sup \rho_ j)(f)=\sup_{j\in J}(\rho_ i(f))\) \(f\in {\mathcal M}^+\). However, there are three possible ways of defining the infimum of \(\{\rho_ j:\) \(j\in J\}\), depending on to which class J the semi-norms \(belong.\) \(f\in M^+\), \(\inf_{\lambda}(\rho_ i)(f)=\inf \{\sum_{J}\rho_ j(f_ j)| f=\sum_{j\in J}f_ j\) a.e., where \(f_ j\in {\mathcal M}^+\) and \(f_ j=0\) for all but countably many j in \(J\}\). \(\inf_{\gamma}(\rho_ j)(f)=\inf \{\sum_{J}\rho_ j(f_ j)| f=\sum_{j\in J}f_ j\) a.e. where \(f_ j\in {\mathcal M}^+\) and \(f_ j=0\) for all but finitely many j's in \(J\}\) \(\inf_{\sigma}(\rho_ j)(f)=\sup(\rho '_ j)'\) where \(\rho '(f)=\sup \{\int_{X}fgd\mu | \rho(g)=1\), \(g\in {\mathcal M}^+\}.\) Let P be the set of all semi-norms an \({\mathcal M}^+\) and R (resp. I,W,S) denote the subset of P consisting of those norms satisfying Riesz-Fisher property (resp. infinite triangle inequality, weak Fatou property, strong Fatou property). Among many other interesting lattice theoretic theorems the authors show that (1) in P, \(\inf_{\sigma}<\inf_{\lambda}<\inf_{\gamma}\) and \(\inf_{\sigma}=(\inf_{\gamma})''=(\inf_{\gamma})''=(\inf_{\sigma}) ''\); (2) If \(\{\rho_ j| j\in J\}\subset I,\) then \(\inf_{\lambda}\) is the right notion of infimum in the respective lattice structure; (3) If \(\{\rho_ j| j\in J\}\subset S\) then \(\inf_{\sigma}\) is the right notion of infimum in the respective lattice structure.