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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?

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    I thank you for your patience. So let me get this straight: ABsolute integrability is sufficient for the equality of double and iterated integrals. BUT AT THE SAME TIME, the absolute integrability is a NECESSARY condition for the equality to hold, simply because a Lebesgue integrable function is always absolutely integrable. If this is what you are saying, then the answer to the OP is a YES but for rather trivial reasons. Is this right?2011-09-29

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One more point:

I had asked this question several months ago and since then I have a much better understanding. My confusion was caused by the Tonelli and Fubini theorems both being called Fubini's theorem. In practice, one uses both of them together.

One can use Tonelli's theorem on $|f|$ to check integrability of a real or complex $f$, since every absolutely integrable function is also integrable. Having established integrability via Tonelli's theorem, one can use Fubini's theorem, which is a version of Tonelli's theorem for general functions. They are so closely related that they are sometimes known simply as the Fubini-Tonelli theorem or even simply as Fubini's theorem. (This is what caused my confusion.)

I am not deleting this question just in case somebody else experiences the same confusion.