Could anyone give me a hand with the following problem?
Let $f$ be a random variable over a probability space $(\Omega,A,\mathbb P)$. Show that $f$ is integrable $\iff $ $\sum\limits_{k=1}^\infty\mathbb P(\{|f|>n\}) < \infty.$
Note: Consider the sets $A_n:=\{n-1<|f|\le n\}$, $n\in \mathbb{N}$ and the random variables $L:=\sum\limits_{k=1}^\infty (n-1)1_{A_n}$ and $R:=\sum\limits_{k=1}^\infty n1_{A_n}$,$n\in\mathbb{N}$.
Hint:
Show that $L\leqslant|f|\leqslant R$ almost surely, and that $R=L+1$. Hence $|f|$ is integrable if and only $L$ is integrable if and only $R$ is integrable. Finally, show that $L=\sum\limits_{n\geqslant1}\mathbf 1_{|f|\gt n}$ hence $E[L]=\sum\limits_{n\geqslant1}P[|f|\gt n]$