# 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).

-

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.