Let $\mathbb{P}$ be a probability measure on $(\Omega,\mathcal{A})$ and $B\in\mathcal{A}$ with $\mathbb{P}(B)>0$. Show that for $\mathbb{P}_B(A)$ $:=\mathbb{P}(A|B)$ holds true $\mathbb{P}_B(A)\ll\mathbb{P}$ and find a density f such that $\mathbb{P}_B(A)=f\mathbb{P}$.
($\ll$ meaning "absolutely continuous with respective to")
I thought about using the Radon-Nikodym-theorem and then f should be the Radon-Nikodym-derivative, but I am uncertain how to obtain f in this case.