Let $f\in L^p$. Show that $f=f\chi_E+g$ where $m(E)<\infty$ and $|g|\le1$. Assume that $m$ is the Lebesgue measure on $\mathbb R$.
Using Chebychev's inequality, I can find a set of finite measure $E$ such that $m(E)=m(x:|f(x)|>1)\le\|f\|_p^p<\infty$. I can't go further.