On approximate derivatives and Krzyzewski-Foran lemma.

Abstract

Let $E \subseteq {\mathbb R}$ and $g:E \to {\mathbb R}$. We show that if $|g(E)| = 0$, then $\underline{g}_{\,ap}(x) \le 0 \le \overline{g}_{ap}(x)$ almost everywhere on $E$, which immediately implies a lemma of Krzyzewski \cite{10} and Foran \cite{7}. The function $g$ is said to satisfy the inverse Lusin condition $(N^{-1})$ on $E$ if $|g^{-1}(H)| = 0$ for every $H \subseteq g(E)$ with $|H| = 0$. We prove that if $g^\prime_{ap}(x)$ exists almost everywhere on $E$, then $g$ is an $N^{-1}$-function if and only if $g_{ap}^\prime(x) \neq 0$ almost everywhere on $E$. We also improve upon Foran's \cite{7} chain rule for approximate derivatives, and obtain necessary and sufficient conditions for its validity almost everywhere.