A product convergence theorem for Henstock--Kurzweil integrals

Abstract

Necessary and sufficient for ${\int }_a^{b}f{g}_n\to {\int }_a^{b}fg$ for all Henstock--Kurzweil integrable functions $f$ is that $g$ be of bounded variation, $g_n$ be uniformly bounded and of uniform bounded variation and, on each compact interval in $(a,b)$, $g_n\to g$ in measure or in the $L^1$ norm. The same conditions are necessary and sufficient for $\Vert f(g_n-g)\Vert \to 0$ for all Henstock--Kurzweil integrable functions $f$. If $g_n\to g$ a.e. then convergence $\Vert f{g}_n\Vert \to \Vert fg\Vert$ for all Henstock--Kurzweil integrable functions $f$ is equivalent to $\Vert f(g_n-g)\Vert \to 0$. This extends a theorem due to Lee Peng-Yee.

See http://www.math.ualberta.ca/~etalvila/research.html. Real. Anal. Exchange (to appear)