Prove that for $X\geq 0$ $\sum_{k=1}^{\infty}P(X>k)\leq E[X]\leq\sum_{k=0}^{\infty}P(X>k)$
Show: $\sum_{k=1}^{\infty}\mu(\left\{f\geq k\right\})\leq\int f\, d\mu\leq\sum_{k=0}^{\infty}\mu(\left\{f>k\right\})$