Home > Radon-Nikodym theorem
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
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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
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where E is the expectation operator. When X is the characteristic function of a set A, one gets the intuitive formula
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The theorem is named for Johann RadonJohann Radon ( December 16, 1887 May 25, 1956) was a mathematician born in Litomerice in Bohemia (now Czech Republic). He wrote a doctoral dissertation on calculus of variations in 1910, at the University of Vienna. He is best known now for: his part in t, who proved the theorem for the special case where the underlying space is in 19131913 is a common year starting on Wednesday. click on link for calendar) Events January-March January 30 House of Lords rejects Irish Home Rule Bill February 1 New York City's Grand Central Station opens as the world's largest train station. February 3 Th, and for Otto Nikodym who proved the general case in 19301930 is the common year starting on Wednesday. see link for calendar) Events January-February January 6 The first diesel-engine automobile trip is completed ( Indianapolis, Indiana, to New York City). January 27 Miguel Primo de Rivera resigns January 30 G.
