I'm working on a problem from Grafakos, Classical Fourier Analysis.
Let $(X, \mu)$ be a measure space and let $E$ be a subset of $X$ with $\mu(E) < \infty$. Assume that $f$ is in $L^{p,\infty}(X,\mu)$ for some $0<p<\infty$.
Show that for $0 < q < p$ we have $$ \int_E |f(x)|^q \, d\mu(x) \le \frac{p}{p-q}\mu(E)^{1-\frac{q}{p}}\|f\|^q_{L^{p,\infty}} $$
The hint is
Use $\mu(E\cap \{|f|>\alpha\}) \le \min(\mu(E),\alpha^{-p}\|f\|^p_{L^{p,\infty}})$.
I can prove the hint, for (1) $E\cap \{|f|>\alpha\} \subset E$, and (2) $\alpha^p d_f(\alpha) \le \|f\|^p_{L^{p,\infty}}$, where $d_f(\alpha) = \mu(|f| > \alpha)$.
But how to apply it to the problem?
After reading the text again, I found the technique already explained in the proof of a proposition (1.1.14).
Let $B = \mu(E)^{-\frac1p}\|f\|_{L^{p,\infty}}$.
$E$ as a measure space itself is $\sigma$-finite. So we have
$$ \begin{align} \int_E |f|^q \, d\mu(x) &= q \int_0^\infty \alpha^{q-1}d_{f|_E}(\alpha) \,d\alpha \\ &\le q\int_0^B \alpha^{q-1}\mu(E)\,d\alpha + q\int_B^\infty\alpha^{q-1}\|f\|^p_{L^{p,\infty}}\alpha^{-p}\,d\alpha \\ &= B^{\,q}\mu(E) - \frac{q}{q-p}B^{\,q-p}\|f\|^p_{L^{p,\infty}} \\ &= \|f\|^q_{L^{p,\infty}}\mu(E)^\frac{p-q}{p} + \frac q{p-q}\|f\|^q_{L^{p,\infty}}\mu(E)^\frac{p-q}p \\ &= \frac p{p-q} \mu(E)^{1-\frac q p}\|f\|^q_{L^{p,\infty}} \end{align} $$
The first equality is proposition 1.1.4.