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The sharp Riesz-type definition for the Henstock-Kurzweil integral. (English) Zbl 1044.26007

Summary: In this paper, we prove that if \(f\) is Henstock-Kurzweil integrable on a compact subinterval \([a,b]\) of the real line, then the following conditions are satisfied: (i) there exists an increasing sequence \(\{X_n\}\) of closed sets whose union is \([a,b]\); (ii) \(\{f\chi_{X_n}\}\) is a sequence of Lebesgue integrable functions on \([a,b]\); (iii) the sequence \(\{f\chi_{X_n}\}\) is Henstock-Kurzweil equi-integrable on \([a,b]\). Subsequently, we deduce that the gauge function in the definition of the Henstock-Kurzweil integral can be chosen to be measurable, and an indefinite Henstock-Kurzweil integral generates a sequence of uniformly absolutely continuous finite variational measures.

MSC:

26A39 Denjoy and Perron integrals, other special integrals
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence