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==Formal definition==
(''This definition may be extended to any probability distribution using the [[measure theory|measure-theoretic]] [[probability axioms|definition of probability]].'')

A [[random variable]] <math>X</math> with values in a [[measurable space]] <math>(\mathcal{X}, \mathcal{A})</math> (usually <math>\mathbb{R}^n</math> with the [[Borel set]]s as measurable subsets) has as [[probability distribution#Formal definition|probability distribution]] the [[pushforward measure]] ''X''<sub>?</sub>''P'' on <math>(\mathcal{X}, \mathcal{A})</math>: the '''density''' of <math>X</math> with respect to a reference measure <math>\mu</math> on <math>(\mathcal{X}, \mathcal{A})</math> is the [[Radon–Nikodym derivative]]:
<math display="block">f = \frac{dX_*P}{d\mu} .</math>

That is, ''f'' is any measurable function with the property that:
<math display="block">\Pr [X \in A ] = \int_{X^{-1} A} \, dP = \int_A f \, d\mu </math>or any measurable set <math>A \in \mathcal{A}.</math>

===Discussion===
In the [[#Absolutely continuous univariate distributions|continuous univariate case above]], the reference measure is the [[Lebesgue measure]]. The [[probability mass function]] of a [[discrete random variable]] is the density with respect to the [[counting measure]] over the sample space (usually the set of [[integer]]s, or some subset thereof).

It is not possible to define a density with reference to an arbitrary measure (e.g. one can not choose the counting measure as a reference for a continuous random variable). Furthermore, when it does exist, the density is almost unique, meaning that any two such densities coincide [[almost everywhere]].