The Tonelli theorem is a fundamental result in measure theory that provides conditions under which the order of integration can be interchanged. Specifically, it applies to non-negative measurable functions, allowing us to switch the order of integration in double integrals without affecting the result. This theorem is particularly useful in simplifying calculations in various fields, including probability and statistics. In addition to its application in double integrals, the Tonelli theorem also lays the groundwork for the Fubini's theorem, which extends the concept to integrable functions. Both theorems are essential in understanding how to handle multiple integrals in mathematical analysis and are widely used in theoretical and applied mathematics.