Mappings into sets of measure zero (Q1084524)

The aim of the paper is to unify several theorems concerning functions satisfying Lusin's property N, absolutely continuous functions, saltus functions and finite Borel measures on [0,1]. These theorems are corollaries from the following general theorem: Let f and g be real valued functions of bounded variation on [0,1]. Let \[ X=\{x\in (0,1): \text{ either } \lim_{h\to 0}| (f(x+h)-f(x))/(g(x+h)-g(x))| = \infty \] \[ \text{ or x lies in the interior of the set }g^{-1}g(x)\}. \] Then a necessary and sufficient condition that \(m(f(X))>0\) (where m stands for Lebesgue outer measure) is that there exists some set \(S\subset [0,1]\) such that \(m(g(S))=0<m(f(S)).\)