In mathematics, the Radon–Nikodym theorem, named after Johann Radon and Otto M. Nikodym, 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
ν ( A ) = ∫ A f d μ , {\displaystyle \nu (A)=\int _{A}f\,d\mu ,}
where ν {\displaystyle \nu } is the new measure being defined for any measurable subset A {\displaystyle A} and the function f {\displaystyle f} is the density at a given point. The integral is with respect to an existing measure μ {\displaystyle \mu } , which may often be the canonical Lebesgue measure on the real line R {\displaystyle \mathbb {R} } or the n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} (corresponding to our standard notions of length, area and volume). For example, if f {\displaystyle f} represented mass density and μ {\displaystyle \mu } was the Lebesgue measure in three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} , then ν ( A ) {\displaystyle \nu (A)} would equal the total mass in a spatial region A {\displaystyle A} . The Radon–Nikodym theorem essentially states that, under certain conditions, any measure ν {\displaystyle \nu } can be expressed in this way with respect to another measure μ {\displaystyle \mu } on the same space. The function f {\displaystyle f} is then called the Radon–Nikodym derivative and is denoted by d ν / d μ {\displaystyle d\nu /d\mu } . An important application is in probability theory, leading to the probability density function of a random variable. The theorem is named after Johann Radon, who proved the theorem for the special case where the underlying space is R n {\displaystyle \mathbb {R} ^{n}} in 1913, and for Otto Nikodym who proved the general case in 1930. In 1936, Hans Freudenthal generalized the Radon–Nikodym theorem by proving the Freudenthal spectral theorem, a result in Riesz space theory; this contains the Radon–Nikodym theorem as a special case. A Banach space Y {\displaystyle Y} is said to have the Radon–Nikodym property if the generalization of the Radon–Nikodym theorem also holds (with the necessary adjustments made) for functions with values in Y {\displaystyle Y} . All Hilbert spaces have the Radon–Nikodym property.
Formal description
Radon–Nikodym theorem The Radon–Nikodym theorem involves a measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} on which two σ-finite measures are defined, μ {\displaystyle \mu } and ν . {\displaystyle \nu .} It states that, if ν ≪ μ {\displaystyle \nu \ll \mu } (that is, if ν {\displaystyle \nu } is absolutely continuous with respect to μ {\displaystyle \mu } ), then there exists a Σ {\displaystyle \Sigma } -measurable function f : X → [ 0 , ∞ ) , {\displaystyle f:X\to [0,\infty ),} such that for any measurable set A ∈ Σ , {\displaystyle A\in \Sigma ,}
ν ( A ) = ∫ A f d μ . {\displaystyle \nu (A)=\int _{A}f\,d\mu .}
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