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Article 42

Thomson's variational measure and nonabsolutely convergent integrals

Real Analysis Exchange 26 (2000/2001), no. 1, 35-50.

Mathematical reviews subject classification: 26A45, 26A39, 26A46, 26A24.

 

Abstract

In 1987 Jarnik and Kurzweil (see J.Jarnik and J. Kurzweil, A general form of the product integral and linear ordinary differential equations, Czech. Math. J.  37 (1987), no. 112, 642-659)  proved the following result:

A function F:[a,b] → R is AC*G on [a,b]  if and only if μF* (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 Article 17.

 

In this paper we shall improve this result, from where we obtain an answer to a question of Faure (see C. A. Faure, A descriptive definition of the KH-Stieltjes integral, Real Analysis Exchange 23 (1997-1998), no. 1, 113-124).

Then using Faure's definition for a Kurzweil-Henstock-Stieltjes integral with respect to a function ω, we give corresponding definitions for:

  • a Denjoy*-Stieltjes integral with respect to ω,
  • a Ward-Perron-Stieltjes integral with respect to ω,
  • a Henstock-Stieltjes variational integral with respect to ω,

and we show that the four integrals are equivalent.

 

Finally, using the notion of AC* with respect to ω (short AC*ω ) and a Lusin condition with respect to ω (short Nω ), we give a Banach-Zarecki type theorem (see S. Saks, Theory of the integral, 2nd. rev. ed., vol. PWN, Monografie Matematyczne, Warsaw, 1937).