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2002-2003 On approximate derivatives and Krzyzewski-Foran lemma.
D. N. Sarkhel
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Real Anal. Exchange 28(1): 175-190 (2002-2003).


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.


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D. N. Sarkhel. "On approximate derivatives and Krzyzewski-Foran lemma.." Real Anal. Exchange 28 (1) 175 - 190, 2002-2003.


Published: 2002-2003
First available in Project Euclid: 12 June 2006

zbMATH: 1059.26004
MathSciNet: MR1973978

Primary: 26A24 , 26A46 , 26A48 , 28A20

Keywords: absolutely continuous. , approximate derivative , chain rule , inverse Lusin condition ($N^{-1}$) , measurable boundaries , monotone (functions)

Rights: Copyright © 2002 Michigan State University Press

Vol.28 • No. 1 • 2002-2003
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