Let $(X, \mu)$ be a measure space which is $\sigma$-finite. $ 1 < p < \infty $. $f : X \to \mathbb C$ is a measurable function. If we know $f$ is in the weak $L_p$ space, i.e. $ ||f||_{L^{(p, \infty)} } : = \sup_{ t > 0} t \lambda_ f (t)^{ 1 / p} \le C $ for some $ C \in \mathbb R$. Where $ \lambda_f (t)$ is the distribution function.
Show that :
There exists $ C' \in \mathbb R$ such that $ | \int_ E f d\mu | \le C'\mu( E)^{ 1/p'}$ for all $E$ with finite measure. Where $ 1/p' = 1 - 1 / p$.
I tried for a long time, but did not make any progress. The statement that I want to show looks quite stronger than my condition. Any help is appreciated.
This question is much easier if you write the weak $L^p$ norm in terms of decreasing rearrangements http://en.wikipedia.org/wiki/Lorentz_space#Decreasing_rearrangements
Then $\int_E f \, d\mu \le \int_0^{[0,\mu(E)]} f^*(s) \, ds \le \|f\|_{p,\infty} \int_0^{[0,\mu(E)]} s^{-1/p} \, ds $.