Integrability in the sense of Cauchy and \(\psi\)-integrability of functions with values in a Banach space (Q5937438)

Let \(X=[A,B]\subset\mathbb R\) and consider finite partitions \(\pi\) of \(X\) consisting of elements of the collection \(\mathcal D\) of sets \(]a,b]\cap X\). If \(\psi\) is a selection map \(\psi:{\mathcal D}\to X\) (\(\psi(D)\in\overline D\) if \(D\neq \emptyset\)), a bounded Banach-valued function \(f:X\to F\) is said to be \(\psi\)-integrable, and that its integral is \(I_\psi(f)=I\), if for each \(\varepsilon >0\) there is a finite partition \(\pi_\varepsilon\) such that \(\|\sum_{D\in\pi} f(D)\mu(D)- I\|_F<\varepsilon\) when \(\pi\) is any partition finer than \(\pi_\varepsilon\). Here \(\mu(D)\) is the length of \(D\). In the main theorem it is shown that, under some technical conditions on \(\psi\), \(\psi\)-integrability of \(f\) implies Riemann integrability (and then \(I_\psi(f)\) coincides with the Riemann integral). Cauchy left-integrability is a special instance of \(\psi\)-integrability, obtained when \(\psi(D)=\inf D\).