Radon–Nikodym theorem

最新推荐文章于 2024-09-06 14:52:18 发布

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最新推荐文章于 2024-09-06 14:52:18 发布

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分类专栏： Martingale+SNC

原文链接：https://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_theorem

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Radon-Nikodym定理是数学中关于测度论的重要结果，它描述了在同一可测空间上定义的两个测度之间的关系。定理表明，在一定条件下，任何测度可以通过另一个已知测度的密度函数来表达。这个密度函数被称为Radon-Nikodym导数，常用于概率论中的概率密度函数。该定理由Johann Radon和Otto Nikodym分别在1913年和1930年证明，并在Riesz空间理论中有更广泛的推广。在Banach空间中，如果满足Radon-Nikodym性质，则可以对函数值进行类似处理。

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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
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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 $ℝ^n$ (corresponding to our standard notions of length, area and volume). For example, if f represented mass density and μ was the Lebesgue measure in three-dimensional space $ℝ^3$ , then ν(A) would equal the total mass in a spatial region A.

The Radon–Nikodym theorem essentially states that, under certain conditions, any measure ν can be expressed in this way with respect to another measure μ on the same space. The function f is then called the Radon–Nikodym derivative and is denoted by $\frac{d\nu}{d\mu}$ . 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 $ℝ^n$ in 1913, and for Otto Nikodym who proved the general case in 1930.[2] 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.[3]

A Banach space Y is said to have the Radon–Nikodym property if the generalization of the Radon–Nikodym theorem also holds, mutatis mutandis, for functions with values in Y. All Hilbert spaces have the Radon–Nikodym property.

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