Is there a Lebesgue integrable function $f$ such that the set of points on no neighborhood of which $f$ is bounded has positive Lebesgue measure?

Question

I tried to prove that there are none using the definition of the Lebesgue integral of nonnegative functions (by converting it into an improper Riemann integral of another variable, see Wikipedia for that) but don't know how to proceed.

Answer 1

Yes. Let $$g(t)=t^{-1/2}\chi_{(0,1)}(t),$$ say $(t_n)$ is a countable dense subset of $\Bbb R$, and define $$f(t)=\sum2^{-n}g(t-t_n).$$

Then $f$ is integrable but $f$ is unbounded in every nonempty open set.