Let $\mathcal{P}$ and $\mathcal{Q}$ be two $\sigma$-finite measures on the measure space $(\Omega,\mathcal{F})$. Let $\mathcal{Q}$ be absolutely continuous with respect to $\mathcal{P}$, that is, for every measurable set $A$, if $\mathcal{P}(A)=0$, then $\mathcal{Q}(A)=0$. According to the Radon-Nikodym theorem, there exists a nonnegative Borel function $f$ such that $\mathcal{Q}(A) = \int_A f \,\text d\mathcal{P}$ for all $A \in \mathcal{F}$. Then, this function $f$ is the Radon-Nikodym derivative of $\mathcal{Q}$ with respect to $\mathcal{P}$ and is denoted by $d\mathcal{Q}/d\mathcal{P}$.
My questions are: i) When will $f=0$? ii) If $\mathcal{P}(A)=\mathcal{Q}(A)=0$, does that implies $f=0$? Or $f$ is an arbitrary number since we have $0/0$?