= Solution
The <layer cake representation>, followed by the assumed tail bound and <Tonelli theorem>, gives
$$
\begin{aligned}
\int_X f_1^p\,d\mu&=p\int_0^\infty a^{p-1}\mu\{f_1>a\}\,da\\
&\leq p\int_0^\infty a^{p-2}\int_{\{g>a\}}g\,d\mu\,da\\
&=p\int_Xg(x)\int_0^{g(x)}a^{p-2}\,da\,d\mu(x)\\
&=\frac p{p-1}\int_Xg^p\,d\mu.
\end{aligned}
$$
All integrands are nonnegative, so this use of <Tonelli theorem> is valid before finiteness of the left-hand side is known. Since $q=p/(p-1)$, the result is
$$
\boxed{\|f_1\|_p^p\leq q\|g\|_p^p}.
$$
Back to article page
