Is an a.e finite function integrable?

Question

It is known that an integrable function is a.e. finite. Is an a.e. finite function integrable? What if the measure is finite?

Answer 1

No. $\frac{1}{x}$ is an example.

Answer 2

No, just consider the constant function $1$. It is not integrable on the real line.

You don't even need an unbounded domain. Let $f(x) = \frac 1x$ and integrate over $[0,1]$ to find a counterexample to your statement.

Answer 3

No. A characteristic function of a non measurable set is everywhere finite, but not integrable.