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

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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]**R. M. McLeod,*The generalized Riemann integral*, Carus Math. Monographs, vol. 20, Math. Assoc. America, Washington D.C., 1980. MR**588510 (82h:26015)****[2]**I. P. Natanson,*Theory of functions of a real variable*. II, transl. by L. F. Boron, Frederick Ungar, New York, 1980. MR**0067952 (16:804c)****[3]**W. F. Pfeffer,*The Riemann-Stieltjes approach to integration*, Technical Report No. 187, National Research Institute for Mathematical Sciences, Pretoria, South Africa.**[4]**S. Saks,*Theory of the integral*, 2nd revised ed., transl. by L. C. Young, Dover, New York, 1964. MR**0167578 (29:4850)****[5]**A. Schurle,*A function is Perron integrable if it has locally small Riemann sums*, J. Austral. Math. Soc. (to appear).

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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