Radon–Nikodym theorem

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

${\displaystyle \nu (A)=\int _{A}f\,d\mu ,}$

where ${\displaystyle \nu }$ is the new measure being defined for any measurable subset ${\displaystyle A}$ and the function ${\displaystyle f}$ is the density at a given point. The integral is with respect to an existing measure ${\displaystyle \mu }$, which may often be the canonical Lebesgue measure on the real line ${\displaystyle \mathbb {R} }$ or the ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ (corresponding to our standard notions of length, area and volume). For example, if ${\displaystyle f}$ represented mass density and ${\displaystyle \mu }$ was the Lebesgue measure in three-dimensional space ${\displaystyle \mathbb {R} ^{3}}$, then ${\displaystyle \nu (A)}$ would equal the total mass in a spatial region ${\displaystyle A}$.

The Radon–Nikodym theorem essentially states that, under certain conditions, any measure ${\displaystyle \nu }$ can be expressed in this way with respect to another measure ${\displaystyle \mu }$ on the same space. The function ${\displaystyle f}$ is then called the Radon–Nikodym derivative and is denoted by ${\displaystyle d\nu /d\mu }$. 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 ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle Y}$ 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 ${\displaystyle Y}$. All Hilbert spaces have the Radon–Nikodym property.