I'm having trouble to prove the equivalence of two (quasi)norms on the weak $L^p$ space.
Assertion: $$\sup_{\mu(A)<\infty}\mu(A)^{(1/p)-1}\int_A |f(x)|dx\leq C\cdot\sup_{t>0}(t\cdot\mu\{|f(x)|>t\}^{1/p}).$$
$\mu$ denotes the Lebesgue measure on $\mathbb{R}^n$. According to a book (Grafakos - Classical Fourier Analysis) $C=p/(p-1)$, but that doesn't help me. I don't know how to handle the integral on the left side, because weak $L^p$-functions aren't necessarily bounded.
I would really appreciate any ideas on how to prove this.
Best regards and sorry for my bad English
If $g$ is a non-negative measurable function, then $$\int_{\mathbb R^n} g(x)\mathrm dx=\int_0^{+\infty}\mu\{g\geqslant t\}\mathrm dt.$$ Using this with $g=|f|\chi_A$, we obtain $$\int_A|f|\mathrm d\mu\leqslant \int_0^{+\infty}\min\{\mu(A),\mu\{|f|\geqslant t\}\}\mathrm dt.$$ Now use the inequality $\mu\{|f|\geqslant t\}\leqslant t^{-p}\sup_{s\geq 0}s^p\mu\{|f|\geqslant s\}$.