On the admissibility of the space \(L_0(\mathcal A,X)\) of vector-valued measurable functions (Q2872181)

Let \(E\) be a Hausdorff topological vector space, and let \(Z\subseteq E\) be a subset of \(E\). If, for any compact subset \(K\) of \(Z\) and any neighbourhood \(V\) of the point \(0\) of \(E\), there is a continuous mapping \(H: K\to V\) such that \(\dim(\text{span\,}H(K))<\infty\) and \(x- H(x)\in V\) for \(x\in K\), then the subset \(Z\) is said to be admissible. If \(Z= E\), then \(E\) is said to be admissible.NEWLINENEWLINE If \(X\) is a Banach space, \(\Omega\) a non-empty set, \({\mathcal A}\) a subalgebra of the power set of \(\Omega\) and \(\mu:{\mathcal A}\to \mathbb R\) is a finitely additive set function, let \(L_0({\mathcal A},X)\) denote the class of \(X\)-valued \(\mu\)-measurable functions on \({\mathcal A}\).NEWLINENEWLINE Statements of the main theorem of this paper indicate that \(L_0({\mathcal A},X)\) is admissible. The introduction of the paper contains statements that the admissibility of other function spaces has been considered by \textit{J. Mach} [``Die Zulässigkeit und gewisse Eigenschaften der Funktionenräume \(L_{\phi,k}\) und \(L_\phi\)'', Ber. Ges. Math. Datenverarbeitung, Bonn 61 (1972; Zbl 0244.46033)] and by \textit{J. Ishii} [J. Fac. Sci., Hokkaido Univ., Ser. I 19, 49--55 (1965; Zbl 0127.32602)].