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I'm not very strong in this subject, so I'm doing some exercises in order to understand it better. But I am currently having problems with this excersize:

We have the target space $(\mathbb R,\mathcal B(\mathbb R),\lambda)$, and $\phi:\mathbb R\to\mathbb R$ is the function given by $\phi(x)=x^2, (x\in\mathbb R)$.
Show, that for every function f in $\mathcal M(\mathcal B(\mathbb R))$:
$f\circ\phi\in\mathcal L^1(\lambda)\Leftarrow\Rightarrow\int_0^\infty{|f(x)|\over\sqrt x}\lambda(dx)\lt \infty$
and
$\int_\mathbb Rf(x^2)\lambda(dx)=\int_o^\infty{f(x) \over\sqrt x}\lambda(dx)$

To be honest, I don't even know how to start it. As I've already said, this is not my strongest subject, so I would really appreciate it if people would explain their answers - but any answers are welcome. Thank you all very much!

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    Do you know the change of variables theorem? Let $y = x^2$ for x > 0 and the result pretty much follows.2012-12-14

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