Radon-Nikodym theorem counting measure absolutely continuous Lebesgue measure probability distribution probability measure
When it is convenient to work with a dominating measure, the Radon-Nikodym theorem is used to define a density as the Radon-Nikodym derivative of the probability distribution of interest with respect to this dominating measure.  Discrete densities are usually defined as this derivative with respect to a counting measure over the set of all possible outcomes.  Densities for absolutely continuous distributions are usually defined as this derivative with respect to the Lebesgue measure.  If a theorem can be proved in this general setting, it holds for both discrete and continuous distributions as well as others;  separate proofs are not required for discrete and continuous distributions.
