Editing Radon–Nikodym theorem (section)

===Probability theory===
The theorem is very important in extending the ideas of [[probability theory]] from probability masses and probability densities defined over real numbers to [[probability measure]]s defined over arbitrary sets. It tells if and how it is possible to change from one probability measure to another. Specifically, the [[probability density function]] of a [[random variable]] is the Radon–Nikodym derivative of the induced measure with respect to some base measure (usually the [[Lebesgue measure]] for [[continuous random variable]]s).

For example, it can be used to prove the existence of [[conditional expectation]] for probability measures. The latter itself is a key concept in [[probability theory]], as [[conditional probability]] is just a special case of it.