Let $p \ge 1$ and $f$ be a Lebesgue measurable function on $\mathbb R$ such that $\int_{\mathbb R} |f(x)|^pdx < \infty$. Show that, $$\int_{\mathbb R} |f(x)|^pdx =\int_0^{\infty} pt^{p-1}\lambda(\{x : |f(x)| > t\})dt$$ where $\lambda$ denotes the Lebesgue measure.
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2https://en.wikipedia.org/wiki/Layer_cake_representation – Maximilian Janisch Sep 14 '20 at 15:24
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See here for the most general case https://math.stackexchange.com/questions/3823829/show-that-int-bbb-r-fxp-dx-int-0-infty-p-tp-1-lambda-lef/3823837#3823837 – Sumanta Sep 14 '20 at 16:15