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Let $f$ be analytic on a domain $\Omega$ of the complex plane, such that the closed disc $\overline{D(0,R)}$ is contained in $\Omega$. What is the difference between $$ \int_{D(0,R)}|f(w)|dm(w)$$ and $$\iint_{D(0,R)}|f(x,y)|dxdy$$ where $dm$ is the Lebesgue measure in $\mathbb R^2=\mathbb C$?

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1  
None. ${}{}{}{}$ –  Giuseppe Negro Jan 3 '13 at 20:08
    
Zip, zilch, nada, nothing. –  JohnD Jan 3 '13 at 20:11
    
understood, thank u all, i'm going to candidate my question to be the most useless in the planet for this year!! :) –  bateman Jan 3 '13 at 20:22
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@bateman: No. It is not useless at all. The Riemann integral and Lebesgue integral are defined very differently, so one needs to prove that they give the same answer when applied to a continuous function. Of course, once you have seen the proof, you can take the result for granted. –  Haskell Curry Jan 3 '13 at 21:06

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