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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.
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence