I have problems computing the decomposition $\nu=\rho+l$ and dervative $\frac{d\mu}{dl}$ in the following cases:
$\nu=\lambda,\space\mu=\delta_{2},$ where $\delta_{2}(x)=1$ if $x=2$ and zero in other case, and $\lambda$ is the Lebesgue measure.
$\nu=\lambda,\space\mu=10\lambda,$ where $\lambda$ is the Lebesgue measure.
$\nu=\delta_{2},\space\mu=\delta_{2}+\delta_{3}.$$
So, we need to use Radon-Nikodym theorem: every measure above is $\sigma-$finite, so for this theorem we can guarantee such decomposition, which is unique. But I can't see how to give explicitly the decomposition and the derivative.
Any kind of help is thanked in advanced