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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A criterion for Perron integrability


Author: D. N. Sarkhel
Journal: Proc. Amer. Math. Soc. 71 (1978), 109-112
MSC: Primary 26A39
MathSciNet review: 0499032
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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\vert{D_ - }u(x) > - \infty ,{D^ - }l(x) < \infty \} $ and $ {E_2} = \{ x\vert{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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0499032-5
PII: S 0002-9939(1978)0499032-5
Keywords: P-integrable, major function, minor function, left major, left minor, right major, right minor, $ V{B_ \ast }$
Article copyright: © Copyright 1978 American Mathematical Society