Do we have $\lVert f\rVert_{L^p(X)}=\int_0^\infty p\alpha^{p-1}d_f(\alpha)\,\mathrm{d}\alpha$ for non $\sigma$-finite $X$?

Question

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?

Answer 1

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