A product convergence theorem for Henstock--Kurzweil integrals

Details A product convergence theorem for Henstock--Kurzweil integrals A product convergence theorem for Henstock--Kurzweil integrals

2003-06-10

arxiv.org/abs/math/0306175v1

Mathematics

Classical Analysis and ODEs

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

Scientific paper

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.

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