What is the Radon-Nikodym derivative?

It is the density function that expresses one measure as an integral against another when the first is absolutely continuous with respect to the second; in probability it is precisely the probability density function.

When can the order of a double integral be swapped?

Tonelli's theorem permits it for non-negative measurable functions on sigma-finite spaces, and Fubini's theorem permits it whenever the function is integrable over the product; together they cover the cases met in practice.

Radon-Nikodym and Product Measures

These results compare and combine measures: the Radon-Nikodym theorem represents one measure as a density times another, while product measures and Fubini's theorem make integration over several variables an iterated process.

Troba un tema amb PaperMindAviatFind papers & topics

Tools & resources

Baixa les diapositives

Learn & explore

VídeoAviat

Definition

The Radon-Nikodym theorem states that a measure absolutely continuous with respect to a sigma-finite measure equals the integral of a density against it; a product measure extends measures on factor spaces to their product so that multi-variable integration can be carried out one variable at a time.

Scope

This topic covers signed and complex measures with the Hahn and Jordan decompositions, absolute continuity and mutual singularity, the Lebesgue decomposition, the Radon-Nikodym theorem and its derivative, the construction of product measures, and the Fubini and Tonelli theorems for interchanging the order of iterated integrals.

Core questions

How is one measure decomposed relative to another into absolutely continuous and singular parts?
When does a measure have a density with respect to another, and what is that density?
How is a measure on a product space built from measures on the factors?
When may the order of an iterated integral be exchanged?

Key theories

Radon-Nikodym theorem: If a measure is absolutely continuous with respect to a sigma-finite measure, it is the integral of a unique density function, the Radon-Nikodym derivative, which is the rigorous foundation of probability densities and conditional expectation.
Fubini-Tonelli theorem: Under sigma-finiteness, an integral over a product space equals either iterated integral, with Tonelli's form for non-negative functions and Fubini's form for integrable ones, justifying the interchange of the order of integration.

Clinical relevance

The Radon-Nikodym derivative is the probability density function and the likelihood ratio of statistics and the rigorous basis of conditional expectation in probability, while product measures and Fubini's theorem underpin the treatment of joint distributions, independence, and multi-dimensional integrals in physics and applied mathematics.

History

Radon proved the density theorem for Euclidean space in 1913 and Nikodym extended it to abstract measures in 1930. Fubini's theorem on iterated integration dates to 1907 and was complemented by Tonelli's non-negative version in 1909, completing the theory of product integration.

Key figures

Johann Radon
Otton Nikodym
Guido Fubini

Seminal works

folland1999
cohn2013

Frequently asked questions

What is the Radon-Nikodym derivative?: It is the density function that expresses one measure as an integral against another when the first is absolutely continuous with respect to the second; in probability it is precisely the probability density function.
When can the order of a double integral be swapped?: Tonelli's theorem permits it for non-negative measurable functions on sigma-finite spaces, and Fubini's theorem permits it whenever the function is integrable over the product; together they cover the cases met in practice.