Theorem

Radon-Nikodym Theorem

Analysis

If one measure is absolutely continuous with respect to another sigma-finite measure, then the first can be written as the integral of a density function, called the Radon-Nikodym derivative, against the second. Named for Johann Radon and Otto Nikodym, it underlies conditional expectation in probability theory.

Facts

Statement

If a measure is absolutely continuous with respect to a second, sigma-finite measure on the same measurable space, there exists a measurable function, the Radon-Nikodym derivative, such that the first measure equals the integral of that function with respect to the second. 1

Proof Year

1930 1

General-case year (Nikodym); Radon proved the R^n special case earlier, in 1913.

Classification

Statement Form

Existence Theorem 1

Statement Form

Identity or Equation 1

Connections

Source Radon-Nikodym Theorem (Wikipedia)

Sources

1. Radon-Nikodym Theorem (Wikipedia)

Wikimedia Foundation

Lead section, opening definition
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.
History section, sentence dating Radon 1913 and Nikodym 1930
The theorem is named after Johann Radon, who proved the theorem for the special case where the underlying space is ℝⁿ in 1913, and for Otto Nikodym who proved the general case in 1930.
Proved By: Johann Radon, Lead section
the Radon-Nikodym theorem, named after Johann Radon

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