Fubini's theorem says that a double integral equals an iterated integral if the double integral is absolutely integrable.
My question is: is the absolute integrability a necessary condition or merely a sufficient condition? To my intuition, a function is not even measurable if it is not absolutely integrable, so I assume that there should be an `only if' part, like this:
A double integral equals an iterated integral if and ONLY IF the double integral is absolutely integrable.
Otherwise, the double integral is not even well defined. Am I missing something?