A new property equivalent to Lebesgue integrability

Author:
Arlo W. Schurle

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 -sets.

**[1]**Robert M. McLeod,*The generalized Riemann integral*, Carus Mathematical Monographs, vol. 20, Mathematical Association of America, Washington, D.C., 1980. MR**588510****[2]**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****[3]**W. F. Pfeffer,*The Riemann-Stieltjes approach to integration*, Technical Report No. 187, National Research Institute for Mathematical Sciences, Pretoria, South Africa.**[4]**Stanisław Saks,*Theory of the integral*, Second revised edition. English translation by L. C. Young. With two additional notes by Stefan Banach, Dover Publications, Inc., New York, 1964. MR**0167578****[5]**A. Schurle,*A function is Perron integrable if it has locally small Riemann sums*, J. Austral. Math. Soc. (to appear).

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
26A39,
26A42

Retrieve articles in all journals with MSC: 26A39, 26A42

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1986-0813820-9

Keywords:
Lebesgue integration,
generalized Riemann integration,
Perron integration

Article copyright:
© Copyright 1986
American Mathematical Society