# Fubini's theorem: if and ONLY IF?

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 think it's also true in cases of nonnegative functions where the integral diverges to $\infty$. – Michael Hardy Sep 29 '11 at 20:01
I suppose that settles the question. So that' why it is not phrased as 'if and only if' ? – Gandhi Viswanathan Sep 29 '11 at 20:07
A function is not even measurable if it is not absolutely integrable: Sorry but this is not so. – Did Sep 29 '11 at 21:01
Agreed, I should not have used the words "not even measurable"... But I am still not clear on the following: can we assign a unique well defined value to the double integral if it is not absolutely integrable? If the positive and negative parts of the integrals are not finite, how can we assign a unique value to the double integral? My doubt is: does Fubini's theorem even make sense when the double integral is not absolutely integrable? – Gandhi Viswanathan Sep 29 '11 at 21:31
@Gandhi: Yes it does. There exist examples of functions of two variables $f(x, y)$ such that both $\int\, dx\int f(x, y)\,dy$ and $\int\, dy\int f(x, y)\,dx$ make sense but are different. Of course in this case $f$ cannot be integrable with respect to $x$ and $y$ jointly. – Giuseppe Negro Sep 29 '11 at 21:47

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.)