Limits and Henstock Integrals of Products

Abstract

When it is known that $\int_a^bf_n\to\int_a^b f$ for a sequence of Henstock integrable functions $\{f_n\}$ we give necessary and sufficient conditions for $\int_a^b f_n\,g_n\to\int_a^b f\,g$ for all convergent sequences $\{g_n\}$ of functions of uniform bounded variation. The conditions are easy to apply and involve either the uniform boundedness or uniform convergence of the indefinite integrals of $f_n$. The proof uses Stieltjes integrals and applies to bounded or unbounded intervals on the real line. It is shown how to define Stieltjes integrals on unbounded intervals without treating them as improper integrals. The special cases $f_n\equiv f$ or $g_n\equiv g$ are also examined. The Abel and Dirichlet tests for integrability of a product are obtained as corollaries as well as a form of the Riemann-Lebesgue lemma. And, if $\Phi : \mathbb{N}\to (0,\infty)$ it is shown what conditions on $\{f_n\}$ and $\{g_n\}$ give $\int_a^b f_n\,g_n=O(\Phi(n))$ as $n\to\infty$.