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Concept# Radon–Nikodym theorem

Summary

In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A measure is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider probability space.
One way to derive a new measure from one already given is to assign a density to each point of the space, then integrate over the measurable subset of interest. This can be expressed as
:\nu(A) = \int_A f , d\mu,
where ν is the new measure being defined for any measurable subset A and the function f is the density at a given point. The integral is with respect to an existing measure μ, which may often be the canonical Lebesgue measure on the real line R or the n-dimensional Euclidean space Rn (corresponding to

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