Real Analysis Exchange

Thomsonʼs Variational Measure and Nonabsolutely Convergent Integrals

Vasile Ene

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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

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

First available in Project Euclid: 2 January 2009

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Zentralblatt MATH identifier

Primary: 26A45: Functions of bounded variation, generalizations 26A39: Denjoy and Perron integrals, other special integrals 26A46: Absolutely continuous functions 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]

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


Ene, Vasile. Thomsonʼs Variational Measure and Nonabsolutely Convergent Integrals. Real Anal. Exchange 26 (2000), no. 1, 35--50.

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  • B. Bongiorno, L. Di Piazza, and V. Skvortsov, A new full descriptive characterization of Denjoy–Perron integral, Real Analysis Exchange 21 (1995-1996), no. 2, 656–663.
  • N. Dunford and J. T. Schwartz, Linear operators, Interscience, 1958.
  • V. Ene, Characterization of $AC^* G \cap {\mathcal C},\; \underline {AC}^* \cap {\mathcal C}_i,\; AC$ and $\underline AC$ functions, Real Analysis Exchange 19 (1994), 491–510.
  • V. Ene, Real functions - current topics,
  • V. Ene, Characterizations of $VB^*G \cap (N)$, Real Analysis Exchange 23 (1997/8), no. 2, 571–600.
  • V. Ene, An elementary proof of the Banach-Zarecki theorem, Real Analysis Exchange 23 (1997/8), no. 1, 295–302.
  • V. Ene, Thomson's variational measure, Real Analysis Exchange 24 (1998/9), no. 2, 523–566.
  • C. A. Faure, Sur le théorème de Denjoy-Young-Saks, C. R. Acad. Sci. Paris 320 (1995), no. Série I, 415–418.
  • C. A. Faure, A descriptive definition of the $KH$-Stieltjes integral, Real Analysis Exchange 23 (1997-1998), no. 1, 113–124.
  • R. Henstock, The general theory of integration, Clarendon Press, Oxford, 1991.
  • J. Jarník and J. Kurzweil, A general form of the product integral and linear ordinary differential equations, Czech. Math. J. 37 (1987), no. 112, 642–659.
  • J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. J. 7 (1957), no. 82, 418–449.
  • P. Y. Lee, On $ACG^*$ functions, Real Analysis Exchange 15 (1989-1990), no. 2, 754–760.
  • I. P. Natanson, Theory of functions of a real variable, 2nd. rev. ed., Ungar, New York, 1961.
  • W. F. Pfeffer, The Riemann approach to integration, Cambridge Univ. Press, New York, 1993.
  • S. Saks, Theory of the integral, 2nd. rev. ed., vol. PWN, Monografie Matematyczne, Warsaw, 1937.
  • D. N. Sarkhel, A wide Perron integral, Bull. Austral. Math. Soc. 34 (1986), 233–251.
  • B. S. Thomson, Real functions, Lect. Notes in Math., vol. 1170, Springer-Verlag, 1985.
  • B. S. Thomson, $\sigma$-finite Borel measures on the real line, Real Analysis Exchange 23 (1997-98), no. 1, 185–192.
  • A. J. Ward, The Perron Stieltjes integral, Math. Zeit. 41 (1936), 578–604.