On Fubini type theorems for the Riemann integral

TL;DR

This paper explores conditions under which Fubini's theorem can be extended to Riemann integrals, enabling repeated integration in multidimensional settings similar to Lebesgue integrals.

Contribution

It establishes a modified Fubini-type theorem for Riemann integrals applicable in important practical cases, bridging a gap with Lebesgue integration.

Findings

Fubini's theorem can be adapted for certain Riemann integrals.

The modified theorem applies to specific multidimensional Riemann integrals.

This approach enhances the evaluation of integrals over complex domains.

Abstract

One of the essential questions of the theory of multidimensional integrals concerns the evaluation of integrals taken in given domains. In the simplest case, when integrating over parallelepipeds, evaluation can easily be performed by repeated integration. In the case of the Lebesgue integral, the question is easily solvable by Fubini's theorem. In the case of the Riemann integral, the situation is complicated by the difference between Jordan and Lebesgue measures. In this paper, we show that in certain important applications of Riemann integrals, one can establish a modification of the theorem on repeated integration in which Fubini's theorem is as powerful as in the case of the Lebesgue integral.