Proving $\int|s|d\mu<\infty \Rightarrow \mu\{x:|s(x)|\geq\epsilon\}<\infty\; \forall \epsilon>0$

Question

How to prove that $$\int|s|d\mu<\infty \Rightarrow \mu\{x:|s(x)|\geq\epsilon\}<\infty\; \forall \epsilon>0?$$

I know that $\int|s|d\mu<\infty$ implies $|s|<\infty$ a.e. Can this fact be used to prove the statement?

Answer 1

Let $\epsilon>0$,then $$\int |s| =\int_{\{|s|\geq \epsilon\}}|s|+\int_{\{|s|< \epsilon\}}|s| \geq \int_{\{|s|\geq \epsilon\}}|s| \geq \epsilon\mu(\{|s| \geq \epsilon\})$$

Thus

$$\mu(\{|s| \geq \epsilon\}) \leq \frac{\int |s|}{\epsilon}$$

So use this inequality on the sets $\{|s| \geq n\}$