The Tonelli-Fubini theory, developed by Leonida Tonelli and Guido Fubini, provides a framework for interchanging the order of integration in multidimensional integrals. Tonelli’s theorem ensures that the integral of a function over a product measure space exists if the integral over each variable exists, while Fubini’s theorem states that the two iterated integrals are equal. The Tonelli-Fubini principle allows for switching the order of integration when certain conditions are met. This theory is fundamental in measure theory and Lebesgue integration, with wide applications in calculus, analysis, and probability theory, enabling the calculation of complex integrals and extending integral concepts to higher dimensions.
