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

[itex]Q(A) = \int_A f \, dP[itex]

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

[itex] E_Q(X) = \int_\Omega X(w)\, dQ = \int_\Omega X(w)\frac{dQ}{dP}\, dP = E_P\left( \frac{dQ}{dP} X \right) [itex]

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

[itex] E_Q(X)=\int_A (dQ)=\int_A\left(\frac{dQ}{dP}\,dP\right). [itex]

The theorem is named for Johann Radon, who proved the theorem for the special case where the underlying space is [itex]R^N[itex] in 1913, and for Otto Nikodym who proved the general case in 1930.de:Satz von Radon-Nikodym

• Art and Cultures
• Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
• Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
• Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
• United States (http://www.academickids.com/encyclopedia/index.php/United_States)
• World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
• Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
• Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
• Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
• Space and Astronomy
• Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)