The variational McShane integral in locally convex spaces
Let \((\Omega, \mathfrak{F},\mu)\) be a \(\sigma\)-finite outer regular quasi-Radon measure space and \(X\) a Hausdorff locally convex linear space. The author studies functions \(f:\Omega\rightarrow X\) which are variationally McShane integrable, i.e., there exists a countably additive set function \(F:\mathfrak{F}\rightarrow X\) with the following property: for any \(\varepsilon >0\) and any continuous seminorm \(p\) on \(X\) there is a gauge \(\Delta\) on \(\Omega\) such that for each partition \(\{(E_i,t_i): i=1,2,\dots\}\) of \(\Omega\) subordinate to \(\Delta\) one has \(\sum_{i=1}^\infty p(\mu(E_i)f(t_i)-F(E_i))<\varepsilon\). Any variationally McShane integrable function is McShane integrable and Pettis integrable. As one of the main results the author gives a precise characterization of variationally McShane integrable functions. This generalizes a result of \textit{L. di Piazza} and \textit{K. Musiał} [Ill. J. Math. 45, 279--289 (2001; Zbl 0999.28006)] on Banach space-valued variationally McShane integrable functions.
scientific article; zbMATH DE number 5652734
scientific article