Let $f:\mathbb R\to [0,\infty]$ be a Lebesgue measurable function which is also Lebesgue integrable. Define, for each positive integer $n$, $S_n=\{x\in \mathbb R:\ f(x)>n\}$. Let $\varepsilon>0$.
I am pondering upon the following:
Does there exist a large enough $N$ such that whenever $n\geq N$, we have $$ \int_{S_n}f\ d\mu < \varepsilon $$
Here's what I am thinking:
Wrtie $X=\mathbb R$. Let $\int_X f\ d\mu =I<\infty$.
Thus we must that $\mu(S_k)<\infty$ for all $k$.
Define $S=\bigcap_{k\in \mathbb N} S_k$.
Since $\mu(S_1)$ is finite, we can write $\lim_{k\to \infty}\mu(S_k)=\mu(S)=\ell$ (say).
Now $\int_Sf\ d\mu\geq n\mu(S)=n\ell$ for all $n$. Thus, if $\ell\neq 0$, then we would have $\int_S f\ d\mu = \infty$, contradicting the integrability of $f$.
So we must have $\ell=0$.
I don't see where to go from here.