Suppose $f$ is a non-negative measurable function on $\mathbb{R}$ such that $\int f<\infty$. Is it true that $m(\{x:f(x)=\infty\})=0?$

Question

Suppose $f$ is a non-negative measurable function on $\mathbb{R}$ such that $\int f<\infty$. Is it true that $m(\{x:f(x)=\infty\})=0?$ Here $m(A)$ denotes the Lebesgue measure of a set $A$.

I think the answer is yes. But my answer is solely intuitive. I tried, but failed to provide a rigorous proof. Could someone please give me a hint? Thanks.

Answer 1

We have $$ \{x:f(x)=\infty\}=\bigcap_{n=1}^\infty\{x:f(x)>n\} $$ and $$ m(\{x:f(x)>n\})=\int_{f(x)>n}dx\le\frac1n\int_{\Bbb R}f(x)\,dx. $$