Completeness of \(L_p\)-spaces in the finitely additive setting and related stories
There are obtained necessary and sufficient conditions for completeness of spaces \(L_p(\Omega,{\mathcal F},\mu)\), \(0\leq p<\infty\), where \(\mu\) is a charge (i.e., finitely additive measure over the field \({\mathcal F}\) of subsets of \(\Omega\)), and convergence almost everywhere is replaces by convergence \(f_n\to f\) hazily, defined by the condition \(\mu^*(\{x:|f_n(x)- f(x)|> \varepsilon\})\to 0\) as \(n\to\infty\) for every \(\varepsilon> 0\), \(\mu^*\) being the ``outer measure'' defined by means of \(\mu\). Among else it is proved that completeness of \(L_0(\Omega,{\mathcal F},\mu)\) is equivalent to completeness of \(L_p(\Omega,{\mathcal F},\mu)\) for any \(p\in [0,\infty)\); there are given several necessary and sufficient conditions for this completeness, also by means of the Stone space of \({\mathcal F}/\mu\). Moreover, there are obtained necessary and sufficient conditions in order that the space \({\mathcal F}/\mu\) with Radon-Nikodým metric \(d_\mu\) be complete. Further, investigations are derived to completeness of \(L_1(\Omega,{\mathcal F},\mu)\).
scientific article; zbMATH DE number 1523557
scientific article