$f$ is finite a.e if it is locally integrable?

Question

I know that when a function $f$ is integrable then it is finite almost every where. I was wondering to know if it is true that whenever $f$ is locally integrable then $f$ is finite almost everywhere. It seems to be true because the book Real Analysis by Stein ,page 107, uses this fact without any proof.

Answer 1

You have not specified the domain. If you are talking about a locally intergable function on $\mathbb R^{n}$ w.r.t. Lebesgue measure then $f$ is integrable on $\{x:\|x\| \leq k\}$ for any positive integer $k$. This implies that $f$ is finite almost everywhere on this ball. Since $\mathbb R^{n}$ is the union of these balls and since a countable union of sets of measure $0$ has measure $0$ it follows that $f$ is finite almost everywhere.

Answer 2

Assume that $f$ is not finite a.e., so there is a subset $B$ of the domain with positive measure on which $f$ is infinity. Then assuming inner regularity of the measure, we can find a compact subset $K$ of $B$ of positive measure, and $\int_K f \ \mathrm d\mu$ would be infinite, contradicting local integrability.