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Recall that for $\sigma$-finite measure spaces $(X,\mathscr{M},\mu)$, for any measurable function $f:X\to\mathbb{C}$, we can recover $\lVert f\rVert_{L^p(X)}$ by defining the distribution function $d_f(\alpha)=\mu(\{\lvert f\rvert>\alpha\})$ and using Fubini-Tonelli to obtain that $$\lVert f\rVert_{L^p(X)}=\int_0^\infty p\alpha^{p-1}d_f(\alpha)\,\mathrm{d}\alpha.$$

Does this remain true if $X$ is no longer $\sigma$-finite?

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Observe that if $f \in L^p(X)$ then the set $\{|f| > 0\}$ is $\sigma$-finite.

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  • $\begingroup$ Suppose that $f$ is not in $L^p$. Then is the R.H.S. infinite? $\endgroup$ – Monstrous Moonshine May 2 '17 at 15:39
  • $\begingroup$ Well, if the R.H.S. is finite, then $\{|f| > 0\}$ is $\sigma$-finite. Does that get you anywhere? $\endgroup$ – Umberto P. May 2 '17 at 15:48
  • $\begingroup$ But $\{\lvert f\rvert>0\}$ can be $\sigma$-finite for $f$ not in $L^p$. $\endgroup$ – Monstrous Moonshine May 2 '17 at 15:50
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    $\begingroup$ Sure it can, but see if you can work it out from here. $\endgroup$ – Umberto P. May 2 '17 at 15:53

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