Lebesgue integral is not injective?

Question

I know that Lebesgue integral is not injective i.e. if

$$ \int\limits_{\Omega} f\, d\mu=\int\limits_{\Omega} g \,d\mu $$

then it is not necessary that $f=g$ on $\Omega$, But is there a simple example that shows this?

Answer 1

How about $f$ is the indicator function of $[0,1]$ and $g$ is the indicator function of $[1,2]$ and $\mu$ is ordinary Lebesgue measure. Then $\int f\,d\mu= \int g\,d\mu$ but $f\ne g$.