Arzela-Ascoli's theorem for Riemann-integrable functions on compact space (Q1083659)

We recall [see \textit{C. Klein} and \textit{S. Rolewicz}, Stud. Math. 80, 109- 118 (1984; Zbl 0577.46040) and \textit{C. Klein}, Lect. Notes Econ. Math. Syst. 26, 382-411 (1984; Zbl 0598.28023)] that the \({\mathbb{R}}^ n\)-valued Riemann integrable functions - resp. more general: the B-valued Darboux integrable functions where B is a Banach space (algebra) - on the compact support K of a non negative Radon measure \(\mu\) form a Banach space (algebra) D(K,\(\mu\),B) with respect to the \(\mu\)-ess sup norm. It is shown, that the Darboux integrable functions with a precompact range also form a Banach space (algebra) CD(K,\(\mu\),B). We prove that CD(K,\(\mu\),B) is isometrically isomorphic to C(Stone(\({\mathfrak A}(K,\mu)),B)\). Here Stone(\({\mathfrak A}(K,\mu))\) is the Stone space of the Boolean algebra \({\mathfrak A}(K,\mu)\) consisting of all \(\mu\)-continuity subsets of K modulo \(\mu\)-content zero sets. C(-,-) denotes the space of continuous functions with respect to the sup norm. We prove that \({\mathfrak M}(B)\times Stone({\mathfrak A}(K,\mu))\) is the maximal ideal space of CD(K,\(\mu\),B). We deduce a direct analogue of Arzela Ascoli's theorem for CD(K,\(\mu\),B).