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?

The function $f(x) = 1$ is finite for all $x\in\Bbb R$ but it's not integrable on $\Bbb R$. — Cameron Williams, Oct 13 '15 at 00:30

score 15 · Answer 1 · answered Oct 13 '15 at 00:30

15

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

answered Oct 13 '15 at 00:30

Paul

score 12 · Answer 2 · answered Oct 13 '15 at 00:29

12

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.

answered Oct 13 '15 at 00:29

Giovanni

score 5 · Answer 3 · answered Oct 13 '15 at 00:28

5

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

answered Oct 13 '15 at 00:28

Joe

3 Answers3