# Are Tonelli's and Fubini's theorem equivalent?

I can derive Fubini's theorem for interated integrals of complex functions from Tonelli's theorem for iterated integrals for unsigned functions. I was wondering whether there is a way to go backwards. I do not think so, because Fubini's theorem assumes the integrals are finite, whereas Tonelli's theorem allows the value of the integral to be $+\infty$. But maybe we can use a limiting argument? This is where I am not clear.

So: is it possible to derive Tonelli's theorem from Fubini's theorem? If so, I would appreciate a proof (or a outline of a proof).

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## 1 Answer

If we have Tonelli's theorem, then by considering positive parts and negative parts separately we immediately obtain Fubini's theorem.

Conversely, assuming Fubini's theorem, Tonelli's theorem follows by monotone convergence argument applied to cut-off functions $f_k(x) = \min \{k, f(x)\} \chi_{B_k}(x)$. You can also find the detail at the Chapter 6.2 of the celebrated textbook Measure and Integral by Wheeden and Zygmund.

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I get it. Very nice. Essentially, this is what I referred to as the "limiting argument." But I had not thought of using cutoff functions... – Gandhi Viswanathan Jun 22 '12 at 19:52
@GandhiViswanathanj : Actually, limiting argument is all that you need, so your idea is both natural and correct. Cutting off blow-ups and tails in such a way that we can apply monotone convergence theorem is just a technical matter. – Sangchul Lee Jun 22 '12 at 19:55