A function that is measurable but not Lebesgue integrable.

Question

There is a theorem in our textbook that says, "Let $f$ be a bounded function on a set of finite measure $E$. Then $f$ is Lebesgue integrable over $E$ if and only if it is measurable."

So I was wondering about an example of a function that was Lebesgue integrable but not measurable. I tried to search for some examples online but couldn't really find anything useful...

Answer 1

The function $1/x$ on $\mathbb{R}$ (defined arbitrarily at $0$) is measurable but it is not Lebesgue integrable. In general, a function is Lebesgue integrable if and only if both the positive part and the negative part of the function has finite Lebesgue integral, which is not true for $1/x$.