arXiv subjects
Publications and source records attributed to Piotr Sworowski.
Given arbitrary $r\ge1$, we construct an HK$_r$-integrable function which is not P$_1$-integrable. This is an extension of Musial et al.\ construction published recently in [Musial, P., Skvortsov, V., Tulone, F.: The HK$_r$-integral is not contained in the P$_r$-integral. Proc. Amer. Math. Soc. {\bf150}(5), 2107--2114 (2022)].
It is proved that any function of a Lusin-type class, the class of $ACG_r$-functions, is differentiable almost everywhere in the sense of a derivative defined in the space~$L^r$, $1\le r<\infty$. This leads to obtaining a full descriptive characterization of a Henstock-Kurzweil-type integral, the $HK_r$-integral, which serves to recover functions from their $L^r$-derivatives. The class $ACG_r$ is compared with the classical Lusin class $ACG$ and it is shown that a continuous $ACG$-function can fail to be $L^r$-differentiable almost everywhere.
We locate Musial\,\&\,Sagher's concept of HK$_r$-inte\-gration within the approximate Henstock--Kurzweil integral theory. If to restrict HK$_r$-integral by requirement that the indefinite HK$_r$-integral is {\em continuous}, then it is included even in the classical Denjoy--Khintchine integral. We provide a direct argument demonstrating that this inclusion is proper.
We characterise the class of uniform limits of functions from Pawlak's class $\mathcal B_1^{**}$. The resulting class $u\mathscr S_1$, which contains functions with the oscillation rank one, is discussed in connection with its linear span. We apply a general, topological space, setting in our discussion.