Let $(X,\mathcal{F},\mu)$ be a finite measure space. If $f:X\rightarrow \mathbb{R}$ is a measurable real function, show that, $f\in L^1(\mu)$ iff $\sum\limits_{n=0}^{\infty}\mu(\{f\geq n\})<\infty$. Am a bit stuck on the $(\Leftarrow)$ direction so any help is appreciated, thanks.
[Math] Condition for integrability on finite measure space
lebesgue-integralmeasure-theory
Best Answer
Consider the sets $A_n=\{|f|\geqslant n\}$ and integrate the double inequality $$ \sum_{n=1}^\infty\mathbf 1_{A_n}\leqslant |f|\leqslant\sum_{n=0}^\infty\mathbf 1_{A_n}. $$ Thus, $f$ is integrable if and only if the series $\sum\limits_n\mu(A_n)$ converges.
The hypothesis that $\mu$ is finite is there to ensure that the $n=0$ term $\mu(A_0)=\mu(X)$ is finite.
The double inequality above stems from the pointwise relations, valid for every nonnegative $t$, $$ \lfloor t\rfloor=\sum_{n=1}^\infty\mathbf 1_{t\geqslant n}\leqslant t\lt1+\lfloor t\rfloor=\sum_{n=0}^\infty\mathbf 1_{t\geqslant n}. $$