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Let $g:\mathbb{R}^n\to\mathbb{R}$ and let $\lambda(\alpha)=m \left\{x:|g(x)|>\alpha\right\}$ If $g\in L^p$. Why $\int_{\mathbb{R}^n}|g(y)|^pdy=-\int_{0}^{\infty} \alpha^p d\lambda(\alpha)$?

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  • $\begingroup$ Are you certain of your equality ? For example the left hand side is $\geq 0$ while the right hand side is $\leq0$. $\endgroup$ – Delta-u Apr 27 at 11:40

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