# Fubini-Tonelli proof purely using complex analysis?

Let $F:[a,b]\times[c,d]\rightarrow \mathbb{R}$ be continuous. Show that:$$\int_{a}^{b}\left(\int_{c}^{d}F(x,y)dy\right)dx = \int_{c}^{d}\left(\int_{a}^{b}F(x,y)dx\right)dy$$

Lebesgue integral hasn't been introduced, and this is from a book concerning complex analysis. Is there a way to show this without the usage of the Lebesgue integral? Using complex analysis? Merci for all hints.

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Yes, there is a version for Riemann-Integrals, too. A quite straight-forward proof can be found in "The College Mathematics Journal" Vol. 33, No. 2 p.126-130, it is available online too.

The main part is a clever application of the mean value theorem. Note also that the term Fubini-Tonelli is mostly used for the Lebesgue-Integral "version" of the theorem.

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Thanks!……………………………. –  VVV Dec 1 '11 at 14:08
No problem :).................... –  Listing Dec 1 '11 at 14:11