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A convergence theorem without pointwise convergence. (English) Zbl 1121.28003
Summary: In [J. Kurzweil, “Henstock-Kurzweil integration: its relation to topological vector spaces” (2000; Zbl 0954.28001)] and also in [J. Kurzweil and J. Jarnik, Bull. Cl. Sci., VI. Sér., Acad. R. Belg. 8, No. 7–12, 217–230 (1997; Zbl 1194.26013)] there is a proof of a convergence theorem without the condition of pointwise convergence of the integrands $$f_n$$. We want to ask whether a similar convergence theorem can be formulated in $$\mathbb{R}^m$$. This paper provides an answer to this question. We will give a convergence theorem in the space of primitives (indefinite integrals) that naturally results in the convergence of the corresponding integrands.
MSC:
 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence