Let $(X, \mathfrak{M}, \mu)$ be a finite measure space, let $\{E_{k}\}_{k=1}^{n}$ a collection of measurable sets, and $\{ c_{k}\}_{k=1}^n$ a collection of real numbers. For $E \in \mathfrak{M}$ define $$\nu(E) = \sum_{k=1}^{n} c_{k} \mu(E \cap E_{k}).$$ Show that $\nu$ is a measure on $(X, \mathfrak{M})$ that is absolutely continuous with respect to $\mu$ and find its Radon-Nikodym derivative $d\nu / d\mu.$
Any hints will be appreciated.