Radon measure

Radon measure. In mathematics, specifically within the branch of measure theory, a Radon measure is a certain well-behaved measure defined on the Borel algebra of a Hausdorff space. The concept, named after the Austrian mathematician Johann Radon, elegantly bridges topology and measure theory by imposing conditions of inner regularity and local finiteness. This framework is fundamental in functional analysis, probability theory, and geometric measure theory, providing the rigorous foundation for integrating functions on locally compact spaces.

Definition and basic properties

A Radon measure on a Hausdorff space is defined as a Borel measure that is both locally finite and inner regular. Local finiteness means every point has a neighborhood (topology) with finite measure, while inner regularity requires the measure of any Borel set to be the supremum of measures of compact subsets contained within it. These properties ensure compatibility between the measure and the underlying topological structure. Important consequences include the fact that on a locally compact space, a Radon measure is also outer regular for all open sets, and every finite measure on a metric space that is also a complete metric space is a Radon measure. The work of Émile Borel and Henri Lebesgue on Borel measure and Lebesgue measure provided essential precursors to this formulation.

Relation to other measures

Radon measures are intimately connected to several other classes of measures in measure theory. Every Radon measure is, by definition, a Borel measure, but the converse requires the additional regularity conditions. On Polish spaces, which include spaces like the real line and Hilbert space, every finite Borel measure is automatically a Radon measure, a result stemming from theorems by Ulam and Andrey Kolmogorov. The relationship with regular measures is particularly close, as Radon measures are precisely the inner regular and locally finite ones. Furthermore, in the context of signed measures and complex measures, one defines signed Radon measure and complex Radon measure by applying the Radon property to their total variation.

Construction and examples

Standard constructions of Radon measures often begin with a positive linear functional on a space of continuous functions, as formalized by the Riesz–Markov–Kakutani representation theorem. A canonical example is the Lebesgue measure on Euclidean space $\backslash mathbb\{R\}^n$, which is a Radon measure. Another primary example is any Dirac measure concentrated at a point in a Hausdorff space. On locally compact groups, the Haar measure, crucial in harmonic analysis, is constructed as a Radon measure. Probability measures on metric spaces, such as those used in the theory of Wiener process, are typically Radon. Product measures formed from Radon measures on σ-compact spaces also yield Radon measures under the product topology.

Riesz representation theorem

The deep connection between Radon measures and continuous linear functionals is captured by the Riesz representation theorem, also known as the Riesz–Markov–Kakutani representation theorem. This cornerstone of functional analysis states that on a locally compact Hausdorff space $X$, every positive linear functional on the space $C\_c(X)$ of continuous functions with compact support can be represented uniquely by a Radon measure. This theorem, developed through the work of Frigyes Riesz, Andrey Markov and Shizuo Kakutani, essentially identifies the dual space of $C\_0(X)$ (functions vanishing at infinity) with the space of finite signed Radon measures. It provides a powerful link between topology, measure theory, and operator theory.

Radon spaces

A topological space is called a Radon space if every finite Borel measure defined on it is a Radon measure. This property ensures that measure and topology interact perfectly. Important classes of spaces are Radon spaces, including all Polish spaces (like the Cantor set and Ba