Real Analysis Exchange

The Hakeʼs Theorem and Variational Measures

Inder K. Rana and Surinder Pal Singh

Full-text: Open access


We give a characterization of the Henstock-Kurzweil integral on \(\R^m\) in terms of variational measures. As an application of this we prove a generalization of the Hake’s theorem to \(\R^m\).

Article information

Real Anal. Exchange, Volume 37, Number 2 (2011), 477-488.

First available in Project Euclid: 15 April 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Primary: 28A50: Integration and disintegration of measures 26A39: Denjoy and Perron integrals, other special integrals
Secondary: 26A39: Denjoy and Perron integrals, other special integrals

Henstock-Kurzweil Integral variational measure charges


Singh, Surinder Pal; Rana, Inder K. The Hakeʼs Theorem and Variational Measures. Real Anal. Exchange 37 (2011), no. 2, 477--488.

Export citation


  • A. M. Bruckner, J. B. Bruckner and B. S. Thomson, Real Analysis, Prentice-Hall, Englewood Cliffs, NJ 1997.
  • C. A. Faure and J. Mawhin, The Hake's property for some integrals over multidimensional intervals, Real Anal. Exchange 20 (1994-95), no. 2, 622–630.
  • C. A. Faure, A descriptive definition of some multidimensional gauge integrals, Czechoslovak Math. J. 45 (1995) 549–562.
  • R. A. Gordon, The integrals of Lebesgue, Denjoy, Perron and Henstock, Amer. Math. Soc., Providence, RI, 1994.
  • P. Y. Lee, Lanzhou Lectures on Henstock Integration, World Scientific, 1989.
  • Tuo-Yeong Lee, A full descriptive definition of the Henstock-Kurzweil integral in the Euclidean space, Proc. London Math. Soc. 87(3) (2003) 677–700.
  • Tuo-Yeong Lee, A measure-theoretic characterization of the Henstock-Kurzweil integral revisited, Czechoslovak Math. J. 58(4) (2008) 1221–1231.
  • Tuo-Yeong Lee, Henstock-Kurzweil integration on Euclidean spaces, Ser. Real Anal. vol. 12, World Scientific Publishing Co., Singapore, 2011.
  • P. Muldowney and V. A. Skvortsov, Improper Riemann integral and the Henstock integral in $R^n$, Math. Notes 78(1-2) (2005) 228–233.
  • L. Di Piazza Variational measures in the theory of the integration in $\R^m$, Czechoslovak Math. J. 51(1) (2001) 95–110.
  • W. F. Pfeffer A descriptive definition of a variational integral and applications, Indiana Univ. Math. J. 40 (1991) 259-270.
  • Stefan Schwabik and Ye Guoju, Topics in Banach Space Integration, Ser. Real Anal. vol. 10, World Scientific Publishing Co., New York, 2005.
  • Stefan Schwabik, Variational measures and the Kurzweil-Henstock integral, Math. Slovaca 59(6) (2009) 731–752.
  • Stefan Schwabik, General integration and extensions I, Czechoslovak Math. J. 60(4) (2010) 961–981.