Let $f\in\mathcal{M}(\mathbb{R})$ non negative. For each $E\subset\mathbb{R}$ measurable we define $\mu_f(E)=\int_{E}f$. Prove
(a) $\mu_f$ is a measure in $\mathcal{M}$
(b) Give an example of a function $f:\mathbb{R}\rightarrow\mathbb{R}$ where $\mu_f$ is complete and another example where $\mu_f$ is not complete.
I've already proved that $\mu_f$ is a measure but I'm having trouble with (b).
$f \equiv 1$ gives an example for which $\mu_f$ is complete, since it coincides with $\mu$.
$f \equiv 0$ gives an example for which $\mu_f$ is not complete. Since $\mu_f(\mathbb{R})=0$, every set would have to be measurable, which we know is not the case.