A criterion for Perron integrability

Abstract

It is shown that a measurable function f : I = [ a , b ] → R e f:I = [a,b] \to {R_e} is necessarily Perron integrable if there exists at least one pair of functions u , l : I → R u,l:I \to R such that (i) u ( x − ) ⩽ u ( x ) ⩽ u ( x + ) u(x - ) \leqslant u(x) \leqslant u(x + ) and l ( x − ) ⩾ l ( x ) ⩾ l ( x + ) l(x - ) \geqslant l(x) \geqslant l(x + ) on I, (ii) I ∖ ( E 1 ∪ E 2 ) I\backslash ({E_1} \cup {E_2}) is countable, where E 1 = { x | D − u ( x ) > − ∞ , D − l ( x ) > ∞ } {E_1} = \{ x|{D_ - }u(x) > - \infty ,{D^ - }l(x) > \infty \} and E 2 = { x | D + u ( x ) > − ∞ , D + l ( x ) > ∞ } {E_2} = \{ x|{D_ + }u(x) > - \infty ,{D^ + }l(x) > \infty \} , and (iii) max { D − u ( x ) , D + u ( x ) } ⩾ f ( x ) ⩾ min { D − l ( x ) , D + l ( x ) } \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.