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2000/2001 Thomsonʼs Variational Measure and Nonabsolutely Convergent Integrals
Vasile Ene
Real Anal. Exchange 26(1): 35-50 (2000/2001).


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.


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Vasile Ene. "Thomsonʼs Variational Measure and Nonabsolutely Convergent Integrals." Real Anal. Exchange 26 (1) 35 - 50, 2000/2001.


Published: 2000/2001
First available in Project Euclid: 2 January 2009

zbMATH: 1010.26008
MathSciNet: MR1825496

Primary: 26A24 , 26A39 , 26A45 , 26A46

Keywords: $AC^*$ , $VB$ , $VB^*$ , $VB^*G$ , $VB^*G$ , the Denjoy$^*$-Stieltjes integral , the Kurzweil-Henstock-Stieltjes integral , Thomson's variational measure

Rights: Copyright © 2000 Michigan State University Press

Vol.26 • No. 1 • 2000/2001
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