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Lusin's theorem for derivatives
with respect to a continuous function


Authors: V. Aversa and D. Preiss
Journal: Proc. Amer. Math. Soc. 127 (1999), 3229-3235
MSC (1991): Primary 26A36; Secondary 26A24, 28E99
DOI: https://doi.org/10.1090/S0002-9939-99-05086-8
Published electronically: May 4, 1999
MathSciNet review: 1636926
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Abstract | References | Similar Articles | Additional Information

Abstract: For a nowhere constant continuous function $g$ on a real interval $I$ and for a Borel measure $\mu$ on $I$, we give simple necessary and sufficient conditions guaranteeing, for any Borel function $f$ on $I$, the existence of a continuous function $F$ on $I$ such that the derivative of $F$ with respect to $g$ is, $\mu$ almost everywhere, equal to $f$.


References [Enhancements On Off] (What's this?)

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  • 2. V. Aversa, Sugli insiemi di peso nullo rispetto ad una funzione continua Ric. Mat. 26, 79-84, 1977. MR 58:1070
  • 3. R. Caccioppoli, L'integrazione e la ricerca delle primitive rispetto ad una funzione continua qualunque. Ann. Mat. Pura e Appl. 40, 15-34, 1955. MR 17:954a
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  • 5. J. Liberman, Théoréme de Denjoy sur la dérivée d'une fonction arbitraire par rapport á une fonction continue, Rec. Mathématique, 221-236, 9(51) 1941. MR 3:74d

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Additional Information

V. Aversa
Affiliation: Dipartimento di Matematica e Statistica, Via Cinzia, 80126 Napoli Italy
Email: aversa@dmsna.dms.unina.it

D. Preiss
Affiliation: Department of Mathematics, University College London, Gower Street, London WCIE-6BT, England
Email: dp@math.ucl.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-99-05086-8
Keywords: Relative derivatives
Received by editor(s): January 22, 1998
Published electronically: May 4, 1999
Additional Notes: The first author was supported by M.P.I fund.
The second author was a visiting Professor at Dipartimento di Matematica e Applicazioni dell’Università di Napoli.
Dedicated: Dedicated to Resia
Communicated by: Frederick W. Gehring
Article copyright: © Copyright 1999 American Mathematical Society

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