Thomsonʼs Variational Measure and Nonabsolutely Convergent Integrals

Abstract

In 1987 Jarník and Kurzweil [11] proved the following result: \emph{A function $F:[a,b] \to {\mathbb R}$ is $AC^*G$ on $[a,b]$ if and only if $\mu_F^*$ (Thomson's variational measure) is absolutely continuous on $[a,b]$ and $F$ is derivable $a.e.$ on $[a,b]$.} But condition $``F$ is derivable $a.e.$ on $[a,b]$'' is superfluous, as it was shown in \cite{Ene19}. In this paper we shall improve this result (from where we obtain an answer to a question of Faure [9]. Then using Faure's definition for a Kurzweil-Henstock-Stieltjes integral with respect to a function $\omega$, we give corresponding definitions for: a Denjoy$^*$-Stieltjes integral with respect to $\omega$, a Ward-Perron-Stieltjes integral with respect to $\omega$, a Henstock-Stieltjes variational integral with respect to $\omega$, and we show that the four integrals are equivalent.