How is this book applying Fubini/Tonelli without assuming $\sigma$-finiteness?

Question

I am reading about $L^p$ spaces on this google book and in proposition 1.1.4 (page 4) it writes $$ p\int_0^\infty \alpha^{p-1} \int_X \chi_{\{x:|f(x)|>\alpha\}}d\mu(x)d\alpha = \int_X \int_0^{|f(x)|}p \alpha^{p-1}d\alpha d\mu(x) $$ stating "we used Fubini's theorem". Every version of Fubini/Tonelli I have ever seen assumes that both measures are $\sigma$-finite, even for nonnegative functions, and as far as I can tell no assumption on the $\sigma$-finiteness of $\mu$ was made here. What version of Fubini is this author applying?

Answer 1

Indeed, we have to work with $\sigma$-finite measures in order to apply Fubini's theorem. Although $\mu$ is not necessarily $\sigma$ finite, the set $\{x;|f(x)|>0\}$ can be written as a countable union of sets of finite measure. Namely, $\{x;|f(x)|>0\}=\bigcup_{n\geqslant 1}\{x,|f(x)|\geqslant n^{—1}\}$, and since $f$ is $\mu$-integrable, $\mu(\{x,|f(x)|\geqslant n^{—1}\})$ is finite. So we can work with $X':=\{x,|f(x)|\geqslant >0\}$ with trace $\sigma$-algebra and measure.