A new property equivalent to Lebesgue integrability
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- by Arlo W. Schurle PDF
- Proc. Amer. Math. Soc. 96 (1986), 103-106 Request permission
Abstract:
Using the generalized Riemann approach to Lebesgue integration we define a new property which is equivalent to Lebesgue integrability for measurable functions. Roughly speaking, this property says that Riemann sums for sufficiently fine partitions of sufficiently small intervals can always be made arbitrarily small. We formulate this property in such a way that it applies to either Lebesgue integration or Perron integration, thus correcting a defect in earlier versions of this idea. The condition of measurability is used only in preliminary results to insure that the support of functions can always be assumed to be ${G_\delta }$-sets.References
- Robert M. McLeod, The generalized Riemann integral, Carus Mathematical Monographs, vol. 20, Mathematical Association of America, Washington, D.C., 1980. MR 588510
- I. P. Natanson, Theory of functions of a real variable, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron with the collaboration of Edwin Hewitt. MR 0067952 W. F. Pfeffer, The Riemann-Stieltjes approach to integration, Technical Report No. 187, National Research Institute for Mathematical Sciences, Pretoria, South Africa.
- 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 A. Schurle, A function is Perron integrable if it has locally small Riemann sums, J. Austral. Math. Soc. (to appear).
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 96 (1986), 103-106
- MSC: Primary 26A39; Secondary 26A42
- DOI: https://doi.org/10.1090/S0002-9939-1986-0813820-9
- MathSciNet review: 813820