Multivariable Integration

This chapter shows that integrating with respect to several variables simultaneously or successively amounts to the same thing, which is the context of Fubini's theorem. It also shows that changing the order of successive integrals of an integrable measure of measures does not affect the result. The chapter focuses on the permutation of variables. It then shows that the permutation of variables does not affect the value of the integral, whether it involves two or more variables. The chapter proves Fubini's theorem by reducing it to the case involving a continuous function and, more precisely, by using the following property of the Cauchy integral. It defines the separation of variables of a measure and shows that the separation of variables is continuous. The chapter expresses that integrating several variables either simultaneously or successively amounts to the same result, which is the content of Fubini's theorem.