Interchanging iterated integration

Abstract:

If $k(x,y)$ is a measurable, real valued, finite a.e. function on $X \times Y$, then necessary and sufficient conditions are given for the two iterated Lebesgue integrals of $k(x,y)$ to be equal and finite by employing Saks’ theorem on the convergence of a sequence of finite measures and the Vitali convergence theorem. The conditions, more general than those of either Fubini’s or Tonelli’s theorems in this case, are applied to an example of a nonintegrable function to show that its iterated integrals are in fact equal and finite.