Real Analysis Exchange

On approximate derivatives and Krzyzewski-Foran lemma.

D. N. Sarkhel

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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.

Article information

Real Anal. Exchange, Volume 28, Number 1 (2002), 175-190.

First available in Project Euclid: 12 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A48: Monotonic functions, generalizations 26A46: Absolutely continuous functions 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

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


Sarkhel, D. N. On approximate derivatives and Krzyzewski-Foran lemma. Real Anal. Exchange 28 (2002), no. 1, 175--190.

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