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Given a measurable function: $f:X\rightarrow C $ on a measured space (X, A, $\mu$)

Show that $\int_X |f|^p d\mu < \infty$ $\leftrightarrow$ $\sum_{n=-\infty}^{\infty} a^n \mu$ ({$(x\in X:a^n \leq |f|^p (x) < a^{n+1}$}) < $\infty $, where $p\in [1,\infty[$, $a\in ]1,\infty[$, by using disjunct decomposition.

I have absolutely no idea, how to approach this problem. For example for the proof of "$\rightarrow$", I don't know, from where I get that given sum..

How do I approach this problem?

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HINT For fixed $a>1$, $f\ne0\iff |f|^p>0\iff \exists n\in\Bbb Z$ such that $a^n\le |f|^p< a^{n+1}$.

Ovbserve that $(0,+\infty)=\bigcup_{n\in\Bbb Z} [a^{n},a^{n+1})$ and the union is disjoint.

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  • $\begingroup$ Where do I consider it exactly? in "$\leftarrow$" or "$\rightarrow$" ? And how do I put them in? $\endgroup$
    – Mathpiano
    Nov 21, 2020 at 16:11
  • $\begingroup$ Both: $\int_X|f|^p=\int_{\{f\ne0\}} |f|^p=\int_{\bigcup A_n}|f|^p=\sum\int_{A_n}|f|^p$. Recall that the union is DISjoiNT. $\endgroup$ Nov 21, 2020 at 16:13
  • $\begingroup$ I'm sorry, but I don't know how to continue or rather how to put the things, you've given as Hint, into $\sum (\int_{A_{n}} |f|^p) = \sum (\int_{X} |f|^p *1_{A_n})$. I think, approaching $|f|^p$ by the limit of a sequence of step-functions $(f_{k})_{k\in N})$ doesn't help here. $\endgroup$
    – Mathpiano
    Nov 21, 2020 at 16:54
  • $\begingroup$ $a^n\mu(A_n)\le \int_{A_n}|f|^p\le a^{n+1}\mu(A_n)$ $\endgroup$ Nov 21, 2020 at 16:58
  • $\begingroup$ I don't get, what I'm gaining from that - another way to say, that I'm stuck. It's up to you, whether you want to help further since you gave that many steps. $\endgroup$
    – Mathpiano
    Nov 21, 2020 at 18:07

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