Abstract
Necessary and sufficient for R b a fgn → R b a fg for all Henstock-Kurzweil inte- grable functions f is that g be of bounded variation, gn be uniformly bounded and of uniform bounded variation and, on each compact interval in (a, b), gn → g in measure or in the L 1 norm. The same conditions are necessary and sufficient fork f(gn − g)k → 0 for all Henstock-Kurzweil integrable functions f. If gn → g a.e. then convergence k fgnk → k fgk for all Henstock-Kurzweil integrable functions f is equivalent to k f(gn − g)k → 0. This extends a theorem due to Lee Peng-Yee. 2000 Mathematics Subject Classification: 26A39, 46E30
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