Tonelli's theorem (functional analysis)

Tonelli's theorem is a result in measure theory that provides conditions under which the order of integration can be interchanged. It is particularly useful in the context of functional analysis and real analysis when dealing with multiple integrals. The theorem typically states the following: Let \( f: X \times Y \to \mathbb{R} \) be a non-negative measurable function defined on the product measure space \( X \) and \( Y \).