A problem in two-dimensional integration

Abstract

AbstractIf two functions of a real variable are integrable over two intervals, say oft, τ, respectively, then the product of the two functions should be integrable over the rectangular product of the two intervals oftand τ. For the Lebesgue integral, definable using non-negative functions alone, the proof is easy. For non-absolute integrals such as the Perron, Çesàro-Perron, and Marcinkiewicz-Zygmund integrals we have difficulties since the functions cannot be assumed non-negative. But the present paper gives a proof.