拉东-尼科迪姆定理

这个定理得名于数学家约翰·拉东以及欧顿·尼科迪姆。拉东在1913年证明了这个定理在背景空间为 R^(N) 时的情况；尼科迪姆则在1930年证明了定理的一般情形。1936年，汉斯·弗洛伊登萨将这个定理推广，证明了里斯空间理论中的弗洛依登萨谱定理。拉东·尼科迪姆定理是后者的一个特例。

拉东-尼科迪姆导数是 f=(dν)/(dμ)

属性

Bermudez et al. (2025) 标准化了以下属性的证明。 λ 几乎处处： (d(ν +μ))/(dλ)=(dν)/(dλ)+(dμ)/(dλ).

若 ν ≪ μ ≪ λ, 则 λ 几乎处处： (dν)/(dλ)=(dν)/(dμ)(dμ)/(dλ).

若 μ ≪ ν 以及 ν ≪ μ, 则 ν 几乎处处： (dμ)/(dν)=((dν)/(dμ))⁻¹.

若 μ ≪ λ 则 ∫ _(X)g dμ =∫ _(X)g(dμ)/(dλ) dλ.

（公式略）

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 dμ,

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 dν /dμ. 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. 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 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. All Hilbert spaces have the Radon–Nikodym property.