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On the characterization of compactness in the space of Henstock-Kurzweil integrable functions. (English) Zbl 1474.26032

Summary: In this paper, we are concerned with a classical question in the space of Henstock-Kurzweil (shortly HK) integrable functions. A negative answer to this question is given by using the theory of the distributional Henstock-Kurzweil (shortly \({D_{HK}}\)) integral. Furthermore, we use convergence to prove a sufficient and necessary condition for a function to be HK integral and then give a characterization of compactness in the space of the HK integrable functions. The results enrich and extend the theory of HK integrable functions space.

MSC:

26A39 Denjoy and Perron integrals, other special integrals
46F10 Operations with distributions and generalized functions
46G12 Measures and integration on abstract linear spaces
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