A product convergence theorem for Henstock--Kurzweil integrals

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 $\mathrm{\parallel }f(g_n-g)\mathrm{\parallel }\to 0$ for all Henstock–Kurzweil integrable functions $f$. If $g_n\to g$ a.e. then convergence $\mathrm{\parallel }f{g}_n\mathrm{\parallel }\to \mathrm{\parallel }fg\mathrm{\parallel }$ for all Henstock–Kurzweil integrable functions $f$ is equivalent to $\mathrm{\parallel }f(g_n-g)\mathrm{\parallel }\to 0$. This extends a theorem due to Lee Peng-Yee.