if $\mu(X)$ is finite and $f$ is finite on X a.e then $\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0$

Question

Let $(X,\mathcal F, \mu)$ be measurable space with $\mu(X)<\infty$. Prove that if function $f$ is measurable and finite on $X$ then $$\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0.$$

I have been asked this question before. However, I missed one condition that $f$ is finite. However, I got a hint without using this hypothesis and I believe that the hint is insufficient. Can anyone give me more detail proof? Thanks

Answer 1

Perhaps, this argument is easier. Just notice the identity $$(|f|=+\infty)=\bigcap_{n\geq 1} (|f|\geq n).$$ Hence $$0=\mu(|f|=+\infty)=\mu(\bigcap_{n\geq 1}(|f|\geq n))=\lim_n\mu(|f|\geq n),$$ where the second equality is continuity from above because $\mu$ is finite.