The pointwise characterization of derivatives of integrals
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- by D. N. Sarkhel PDF
- Proc. Amer. Math. Soc. 63 (1977), 125-128 Request permission
Abstract:
Let $\sigma$ denote the indefinite integral of a summable function f on ${R_n}$. We give necessary and sufficient conditions for $\sigma$ to possess a strong, or ordinary, or general derivative, equal to $f(x)$ at a point x of approximate continuity of f. Munroe [3] states that when f is a bounded function then $f(x)$ is the strong derivative of $\sigma$ at x if and only if x is a point of approximate continuity of f. We point out an error in the proof of the only if part of this result and show by example that this part of Munroe’s result is in fact false.References
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H. Busemann and W. Feller, Zur Differentiation der Lebesgueschen Integrale, Fund. Math. 22 (1934), 226-256.
B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Fund. Math. 25 (1935),217-234.
- M. E. Munroe, Introduction to measure and integration, Addison-Wesley Publishing Co., Inc., Cambridge, Mass., 1953. MR 0053186
- Stanisław Saks, Theory of the integral, Second revised edition, Dover Publications, Inc., New York, 1964. English translation by L. C. Young; With two additional notes by Stefan Banach. MR 0167578
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 63 (1977), 125-128
- MSC: Primary 28A15
- DOI: https://doi.org/10.1090/S0002-9939-1977-0437703-6
- MathSciNet review: 0437703