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
- <math>Q(A) = \int_A f \, dP<math>
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. A similar theorem can be proven for signed measure.
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
- <math> E_Q(X) = \int_\Omega X(w)\, dQ = \int_\Omega X(w)\frac{dQ}{dP}\, dP = E_P\left( \frac{dQ}{dP} X \right) <math>
where E is the expectation operator. When X is the characteristic function of a set A, one gets the intuitive formula
- <math> E_Q(X)=\int_A (dQ)=\int_A\left(\frac{dQ}{dP}\,dP\right). <math>
The theorem is named for Johann Radon, who proved the theorem for the special case where the underlying space is <math>R^N<math> in 1913, and for Otto Nikodym who proved the general case in 1930.de:Satz von Radon-Nikodym