A compactness criterion and the Hausdorff measure of noncompactness for subsets of the space of measurable functions (Q1083649)

Let \(\Omega\) be a Lebesgue-measurable subset of \({\mathbb{R}}^ n\), M(\(\Omega)\) the space of all Lebesgue-measurable functions on \(\Omega\) to \({\mathbb{R}}\) and \(T_ 0(\Omega)\) its subspace of all totally measurable functions [in the sense of \textit{N. Dunford} and \textit{J. T. Schwartz}, Linear operators, Part I (1958; Zbl 0084.104) definition III.2.10]. The authors study compactness in M(\(\Omega)\) and the measure of noncompactness in \(T_ 0(\Omega)\). The results are related to the compactness criterion of Fréchet-Šmulian, s. Theorem IV.11.1 of the above cited book.