  Necessary and sufficient for $\int_a^bfg_n\to \int_a^bfg$ 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 $\|f(g_n-g)\|\to 0$ for
all Henstock--Kurzweil integrable functions $f$. If $g_n\to g$ a.e. then
convergence $\|fg_n\|\to\|fg\|$ for all Henstock--Kurzweil integrable functions
$f$ is equivalent to $\|f(g_n-g)\|\to 0$. This extends a theorem due to Lee
Peng-Yee.
