Radon–Nikodym theorem

Radon–Nikodym theorem
Name	Radon–Nikodym theorem
Field	Measure theory
First proof by	Johann Radon, Otto Nikodym
First proof date	1913, 1930
Generalizations	Lebesgue decomposition theorem

Contents

Statement of the theorem
Formal proof sketch
Properties and consequences
Applications
Examples
Extensions and generalizations

Radon–Nikodym theorem. In measure theory, a fundamental branch of mathematical analysis, the Radon–Nikodym theorem provides conditions under which one σ-finite measure can be expressed as an integral with respect to another. It establishes the existence of a measurable function, known as the Radon–Nikodym derivative, which acts as a density. This result is pivotal in probability theory, functional analysis, and mathematical physics, forming the theoretical backbone for concepts like conditional expectation and change of measure.

Statement of the theorem

Let $(X, \mathcal{F})$ be a measurable space. Consider two σ-finite measures $\mu$ and $\nu$ defined on this space. If $\nu$ is absolutely continuous with respect to $\mu$ (denoted $\nu \ll \mu$ ), meaning $\mu(A)=0$ implies $\nu(A)=0$ for all $A \in \mathcal{F}$ , then there exists a measurable function $f: X \to [0, \infty)$ such that for every measurable set $A$ , $\nu(A) = \int_A f \, d\mu$ . The function $f$ is unique up to $\mu$ -almost everywhere equality and is denoted $\tfrac{d\nu}{d\mu}$ . This formulation is central to the Lebesgue decomposition theorem, which separates a measure into absolutely continuous and singular measure parts relative to another.

Formal proof sketch

The standard proof leverages the Hahn decomposition theorem applied to the signed measure $\nu - \alpha \mu$ for positive scalars $\alpha$ . A key step involves constructing the function $f$ as a supremum over integrals of simple functions, utilizing properties of the Lebesgue integral. The argument often proceeds by first proving the theorem for finite measures, using a maximal function argument inspired by the Hardy–Littlewood maximal function, and then extending to the σ-finite case via a countable partition of the space $X$ . The uniqueness follows from the properties of integration and the definition of almost everywhere.

Properties and consequences

The Radon–Nikodym derivative $\tfrac{d\nu}{d\mu}$ satisfies a chain rule: if $\lambda \ll \nu \ll \mu$ , then $\tfrac{d\lambda}{d\mu} = \tfrac{d\lambda}{d\nu} \cdot \tfrac{d\nu}{d\mu}$ $\mu$ -almost everywhere. In probability theory, when $\mu$ is a probability measure and $\nu$ is another measure, the derivative provides the likelihood ratio. The theorem is foundational for the Riesz representation theorem for the dual of <math>L^p</math> spaces, identifying continuous linear functionals with integrals. It also underpins the Girsanov theorem in stochastic calculus, which is crucial for financial mathematics.

Applications

In probability theory, the theorem is essential for defining conditional expectation given a σ-algebra, as developed by Andrey Kolmogorov. It facilitates the change of measure technique used in derivative pricing models like the Black–Scholes model. Within statistics, it justifies the existence of probability density functions for continuous random variables. In quantum mechanics, the formalism of density operators relies on analogous concepts. The theorem also appears in ergodic theory, particularly in the study of invariant measures for dynamical systems, and in information theory for defining Kullback–Leibler divergence.

Examples

A canonical example is on the real line with the Borel σ-algebra. Let $\mu$ be the Lebesgue measure and $\nu$ be a measure defined by $\nu(A) = \int_A g(x) \, dx$ for some integrable function $g$ , such as a probability density function like the normal distribution. Then $\tfrac{d\nu}{d\mu} = g$ . In a discrete setting, if $\mu$ \nu is any measure, the derivative is simply the mass function. For a singular example, the Cantor distribution relative to Lebesgue measure has no Radon–Nikodym derivative, as it is not absolutely continuous.

Extensions and generalizations

The theorem extends to signed measures and complex measures via the Hahn decomposition theorem. A significant generalization is the Lebesgue decomposition theorem, which decomposes any σ-finite measure into absolutely continuous and singular parts relative to another. In functional analysis, the concept is abstracted in the theory of vector measures and the Bochner integral. For finitely additive measures, an analogous result is given by the Yosida–Hewitt decomposition. In non-commutative geometry, versions exist for von Neumann algebras, related to the Tomita–Takesaki theory. The Krein–Milman theorem and Choquet theory also provide related integral representation results in convex analysis.

Category:Measure theory Category:Theorems in analysis Category:Mathematical theorems