# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## A criterion for Perron integrabilityHTML articles powered by AMS MathViewer

by D. N. Sarkhel
Proc. Amer. Math. Soc. 71 (1978), 109-112 Request permission

## Abstract:

It is shown that a measurable function $f:I = [a,b] \to {R_e}$ is necessarily Perron integrable if there exists at least one pair of functions $u,l:I \to R$ such that (i) $u(x - ) \leqslant u(x) \leqslant u(x + )$ and $l(x - ) \geqslant l(x) \geqslant l(x + )$ on I, (ii) $I\backslash ({E_1} \cup {E_2})$ is countable, where ${E_1} = \{ x|{D_ - }u(x) > - \infty ,{D^ - }l(x) < \infty \}$ and ${E_2} = \{ x|{D_ + }u(x) > - \infty ,{D^ + }l(x) < \infty \}$, and (iii) $\max \{ {D_ - }u(x),{D_ + }u(x)\} \geqslant f(x) \geqslant \min \{ {D^ - }l(x),{D^ + }l(x)\}$ a.e. on I. In the special case when u and l are respectively major and minor functions of f in the sense of H. Bauer, the result was proved by J. Marcinkiewicz.
References
• Hans Bauer, Der Perronsche Integralbegriff und seine Beziehung zum Lebesgueschen, Monatsh. Math. Phys. 26 (1915), no. 1, 153–198 (German). MR 1548647, DOI 10.1007/BF01999447
• Heinrich Hake, Über de la Vallée Poussins Ober-und Unterfunktionen einfacher Integrale und die Integraldefinition von Perron, Math. Ann. 83 (1921), no. 1-2, 119–142 (German). MR 1512004, DOI 10.1007/BF01464233
• E. J. McShane, Integration, Princeton Math. Series, 7, Princeton Univ. Press, Princeton, N. J., 1944.
• 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
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