Variational Henstock integrability of Banach space valued functions. (Q2810147)

In this paper, the authors focus on variational Henstock integrability of Banach space valued strongly measurable functions \(f:[0,1]\rightarrow X\) of the form NEWLINE\[NEWLINEf=\sum_{n=1}^{\infty}x_n\chi_{E_n},NEWLINE\]NEWLINE where \(x_n\in X\), \(E_n\subset [a,b]\) are measurable and pairwise disjoint.NEWLINENEWLINE A function \(f:[0,1]\rightarrow X\) is said to be \textit{variationally Henstock integrable} (briefly \(\text{vH}\)-\textit{integrable}) on \([0,1]\), if there exists an additive function \(F:\mathcal{I}\rightarrow X\), \(\mathcal{I}\) denotes the family of all closed subintervals of \([0,1]\), satisfying the following condition:NEWLINENEWLINE Given \(\varepsilon >0\), there exists a gauge \(\delta\) such that if \(P=\{ (I_i,t_i):i=1,\dots ,p\}\) is a \(\delta\)-fine partition in \([0,1]\), then NEWLINE\[NEWLINE\sum_{i=1}^p\| f(t_i)|I_i|-F(I_i)\| <\varepsilon.NEWLINE\]NEWLINE In integration theory, the integrability of \(\sum_{n=1}^{\infty}x_n\chi_{E_n}\) is related to the convergence of the series \(\sum_{n=1}^{\infty}x_n|E_n|\), e.g., absolute convergence of the series \(\sum_{n=1}^{\infty}x_n|E_n|\) is a necessary and sufficient condition for Bochner integrability of \(\sum_{n=1}^{\infty}x_n\chi_{E_n}\), while unconditional convergence is a necessary and sufficient condition for Pettis integrability. However, in general, the series \(\sum_{n=1}^{\infty}x_n|E_n|\) is only conditionally convergent in the case of Kurzweil-Henstock or variational Henstock integrals.NEWLINENEWLINE In this paper, the authors prove that \(f\) is \(\text{vH}\)-integrable if and only if the series \(\sum_{n=1}^{\infty}x_n|E_n|\) is convergent, where \(E_n\subset [a_{n+1},a_n)\) is Lebesgue measurable, and \(\{ a_n\}\) is a decreasing sequence converging to zero with \(a_1=1\). Moreover, NEWLINE\[NEWLINE(\text{vH})\int_If=\sum_{n=1}^{\infty}x_n|E_n\cap I|,\quad\forall I\in \mathcal{I}NEWLINE\]NEWLINE and the series \(\sum_{n=1}^{\infty}x_n|E_n\cap I|\) is uniformly convergent on \(\mathcal{I}\).NEWLINENEWLINE The authors also give a necessary and sufficient condition (related to a particular order of the sets \(E_n\)) for the \(\text{vH}\)-integrability of a special type of such functions, which is obtained as a corollary of the previous conclusion.