Suppose that $f$ is a Lebesgue measurable function and it is Lebesgue integrable on a compact set $C$ which is a subset of $\mathbb{R^n}$. Then my question is, does $f$ have to be bounded on $C$?
This is a true property of continuous functions, but of course a Lebesgue integrable function need not be continuous.
Best Answer
No, of course: consider $f(x)=\begin{cases}x^{-1/2}&\text{if }x\in(0,1]\\0&\text{if }x=0\end{cases}$ on $[0,1]$.