Calculating Radon Nikodym derivative

Yes, the Radon-Nikodym derivative is $1_{X\setminus\{0\}}$. Note that $1_{X\setminus\{0\}}\ne1$ ($\mu$-a.e.) as $0$ is an atom for $\mu$.

The Radon-Nikodym derivative is $1_{X\setminus\{0\}}$. Note that $1_{X\setminus\{0\}}\ne1$ ($\mu$-a.e.) as $0$ is an atom for $\mu$. Add: Notice that $1_{X}$ is wrong, because μ({0})==1!=0, the R-N derivative is unique in the sense of μ-a.e.