×

Spaces of measurable functions. I. (English) Zbl 0754.46020

Summary: Given a set \(\Omega\neq\emptyset\), an algebra \({\mathcal A}\) on \(\Omega\) and a submeasure \(\eta\) on the power set of \(\Omega\), the purpose of this note is to find some conditions on \(\eta\) so that the functions of \(L_ 0({\mathcal A},\eta)\) are \({\mathcal A}\)-measurable in the sense of P. de Lucia and H. Weber [Completeness of function spaces, Ric. Mat., to appear] and to study the relation between properties of \(\eta\) on the closure of \({\mathcal A}\) in the \(\eta\)-topology and properties of \(L_ 0\) such as \(\sigma\)-Fatou property and pseudo \(\sigma\)-Lebesgue property.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
PDFBibTeX XMLCite