Lebesgue integral of non-zero function

Question

Let's say that $f$ is a characteristic function such that $f \neq 0$. I have a certain measure $\mu$. Can I conclude that the Lebesgue integral $\int f d\mu \neq 0$ and why or why not?

Answer 1

You can't. If $C$ is the cantor set, $f$ is the characteristic function for $C$ and $\mu$ is the Lebesgue measure on $[0,1]$ then $$\int f dx = \mu(C)=0$$