## Real Analysis Exchange

### Thomsonʼs Variational Measure and Nonabsolutely Convergent Integrals

Vasile Ene

#### 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.

#### Article information

Source
Real Anal. Exchange, Volume 26, Number 1 (2000), 35-50.

Dates
First available in Project Euclid: 2 January 2009

https://projecteuclid.org/euclid.rae/1230939146

Mathematical Reviews number (MathSciNet)
MR1825496

Zentralblatt MATH identifier
1010.26008

#### Citation

Ene, Vasile. Thomsonʼs Variational Measure and Nonabsolutely Convergent Integrals. Real Anal. Exchange 26 (2000), no. 1, 35--50. https://projecteuclid.org/euclid.rae/1230939146

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