Chapter 13 - The Henstock—Kurzweil Integral

Abstract

This chapter reviews Henstock–Kurzweil integral. The method of Kurzweil and Henstock is based on the notions of gauge Sind partition. The generalized variation to give a descriptive definition of the Denjoy–Perron integral is presented in the chapter. It is possible to formulate a third characterization of Henstock–Kurzweil primitives in terms of the absolute continuity of a useful extension of the classical notion of the variation of a function. A convergence theorem that includes the monotone convergence theorem and the dominated convergence theorem is an extension of the Vitali's theorem called controlled convergence theorem. It is proved in the chapter that the notions of equiconvergent sequence and control convergent sequence are equivalent. The integrals induced by differentiation bases are analyzed in the chapter. The examples of such integrals are the dyadic integral, the approximate integral, the symmetric integral, and the approximate symmetric integral. It is found that the problem of recovering an approximately continuous function from its approximate derivative is solved by the approximate integral.