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I am looking at an exercise (6.2) in Kesavan's book on Functional Analysis. It says: Let $(X, S, \mu)$ be a measure space and let $1 \leq p < \infty$. Define, for $t>0$, $$ h_f(t) = \mu(\{|f| > t\}. $$ Show that $$ ||f||_p^p = p \int_0^\infty t^{p-1} h_f(t) dt. $$ The book recommends using Fubini's Theorem.

I have been stuck for quite a while. I thought I would simply start with the $L^1$-case which would imply we need to show that $$ ||f||_1 = \int_0^\infty \int_X \chi_{\{|f|>t\}}(t) d\mu \; dt \\ = \int_X \int_0^\infty \chi_{\{|f|>t\}}(t) dt \; d\mu. $$ So there would be the use case for Fubini. I have no clue how to show it. Any help would be appreciated. Thanks

Carlos Esparza
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Fred Kel
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1 Answers1

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Note that $$\int_0^\infty \chi_{\{| f(x) | > t\}} dt = \int_0^{| f(x)|} dt = |f(x)|.$$

coudy
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