Limits and Henstock integrals of products. (English) Zbl 1014.26014

Talvila, Erik

Real Anal. Exch. 25(1999-2000), No. 2, 907-918 (2000).

The paper gives necessary and sufficient conditions for a sequence \((f_n)\) of Henstock-Kurzweil integrable functions such that \(\int_a^b f_n \to \int_a^b f,\) in order that \(\int_a^b f_ng_n \to \int_a^b fg,\) for all convergent sequences \((g_n)\) of functions of uniform bounded variation. As corollaries, the author obtains Abel-Dirichlet-type tests for integrability of a product and a form of the Riemann-Lebesgue lemma. The proofs make use of Riemann-Stieltjes integrals.

Reviewer: Jean Mawhin (Louvain-la-Neuve)

MSC:

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

Keywords:

Henstock-Kurzweil integral; Abel-Dirichlet-type tests