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A criterion for Perron integrability


Author: D. N. Sarkhel
Journal: Proc. Amer. Math. Soc. 71 (1978), 109-112
MSC: Primary 26A39
DOI: https://doi.org/10.1090/S0002-9939-1978-0499032-5
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 ${E_1} = \{ x|{D_ - }u(x) > - \infty ,{D^ - }l(x) < \infty \}$ 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 [Enhancements On Off] (What's this?)

  • Hans Bauer, Der Perronsche Integralbegriff und seine Beziehung zum Lebesgueschen, Monatsh. Math. Phys. 26 (1915), no. 1, 153–198 (German). MR 1548647, DOI https://doi.org/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 https://doi.org/10.1007/BF01464233
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Keywords: P-integrable, major function, minor function, left major, left minor, right major, right minor, <!– MATH $V{B_ \ast }$ –> <IMG WIDTH="46" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$V{B_ \ast }$">
Article copyright: © Copyright 1978 American Mathematical Society