Radon-Nikodym theorem

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In mathematics, the Radon-Nikodym theorem is a result in functional analysis that states that if a measure Q is absolutely continuous with respect to another sigma-finite measure P then there is a measurable function f, taking values in [0,∞], on the underlying space such that

for any measurable set A.

The function f is defined up to a null set, that is: if g satisfies the same property, then f=g almost everywhere. It is commonly written dQ/dP and is called the Radon-Nikodym derivative. The choice of notation and the name of the function reflects the fact that the function is analogous to a derivative in calculus in the sense that it describes the rate of change of density of one measure with respect to another.

It follows trivially from the definition of the derivative that, when P and Q are probability measures over the probability space Ω, and X is a random variable then

where E is the expectation operator. When X is the characteristic function of a set A, one gets the intuitive formula

The theorem is named for Johann Radon, who proved the theorem for the special case where the underlying space is in 1913, and for Otto Nikodym who proved the general case in 1930.