A criterion for Perron integrability

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.