On the Cartesian product of two measure spaces one can construct a measure—hence, an integral—which is directly connected with the measure on each factor. Then, the natural problem that arises is how to reduce a double (or multiple) integral to the computation of two (or more) simple integrals. Such a question plays a crucial role in Lebesgue integration.
Cannarsa, P., & D’Aprile, T. (2015). Product measures. In UNITEXT - La Matematica per il 3 piu 2 (Vol. 89, pp. 107–130). Springer-Verlag Italia s.r.l. https://doi.org/10.1007/978-3-319-17019-0_4