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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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. Vincenzo Aversa, Sugli insiemi di peso nullo rispetto ad una funzione continua, Ricerche Mat. 26 (1977), no. 1, 79–84 (Italian). MR 0480922 (58 #1070)
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  • 4. Mihai Grădinaru, On the derivative with respect to a function with applications to Riemann-Stieltjes integral, Seminar on Mathematical Analysis (Cluj-Napoca, 1989–1990) Preprint, vol. 90, “Babeş-Bolyai” Univ., Cluj-Napoca, 1990, pp. 21–28. MR 1168047 (93g:26008)
  • 5. Joseph Liberman, Théorème de Denjoy sur la dérivée d’une fonction arbitraire par rapport à une fonction continue, Rec. Math. [Mat. Sbornik] N. S. 9 (51) (1941), 221–236 (French., with Russian summary). MR 0004874 (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: http://dx.doi.org/10.1090/S0002-9939-99-05086-8
PII: S 0002-9939(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