Variation of \(f\) on \(E\) and Lebesgue outer measure of \(fE\) (Q1175321)

The author uses his definition of a differential based upon a Kurzweil- Henstock integral to prove that if \(f\) is a real-valued function on \(K=[a,b]\) and \(E\subset K\) is \(df\)-null (i.e., the upper integral \(\overline\int_ K1_ E| df|=0\)), then \(f(E)\) is Lebesgue-null. If \(f\) is continuous and of bounded variation, the converse is true. Consequently, for such a function, Lusin's condition (\(N\)) that \(f\) maps Lebesgue null-sets into Lebesgue-Null sets is nothing but the absolute continuity condition that every Lebesgue-null set is \(df\)-null.