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3 1 I need some help with the following: Suppose that $T:[0,1] rightarrow [0,1]$ is an homemorphism, which satisfies that $T(0)=0$ . It is really intuitive that $$int_0^1 |T(y) - y| dy = int_0^1 |T^{-1}(y) - y| dy$$ since $T$ and $T^{-1}$ are symmetric with respect to the identity, and so the area between $T$ and identity will be the same as $T^{-1}$ and identity, which is the equality posted above. For me, it's intuitive but I can't get a proof of that. Also, what happend if we change the absolute value by square (change the $L_1$ norm for $L_2$ norm). Thanks a lot! calculus real-analysis analysis measure-theory lebesgue-integral share | cite | improve this question