I can't understand the answer of this question. More precisely, I can't understand this step:
$$\int_B |u-u_B|^p \ dx=p\int_0^\infty t^{p-1}|\{x\in B: |u(x)-u_B|>t\}|\ dt $$
Any help will be very appreciated. Thanks!
BMO functions are $L^p_{loc}$
1
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real-analysis
functional-analysis
measure-theory
1 Answers
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This is an application of Tonelli's Theorem.
Indeed we have
\begin{align} p \int_0^{\infty} t^{p-1}|\{x \in B : |u - u_B| > t\}|\,dt = &\ p\int_0^{\infty}t^{p-1}\int_B\chi_{\{|u - u_B| > t\}}(x)\,dxdt \\ = &\ p \int_B\int_0^{|u-u_B|}t^{p-1}\,dtdx \\ = &\ \int_B|u - u_B|^p\,dx. \end{align}