Dual formulation of weak $L^p$

Question

Let $1<p\leq \infty$. Then we have$$||f||_{L^{p, \infty}(X,d\mu)} \sim_psup\{\mu(E)^\frac{-1}{p'}|\int_E f d\mu|:0<\mu(E)<\infty\}$$ Where$||f||_{L^{p, \infty}(X,d\mu)} = \sup_{\lambda\in\left(0, \infty\right)} \lambda (\mu(\{|f|>\lambda\}))^\frac{1}{p}$ and $\frac{1}{p}+\frac{1}{p'}=1$
$A\sim_pB$ means $\exists C_p, D_p $ s.t. $A<C_pB$ and $B<D_pA$

I have no idea in proving both direction.