Radon measure

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In mathematics, a Radon measure, named after Johann Radon, on a Hausdorff topological space X is defined in measure theory to be a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular.

1 Motivation
2 Definitions
3 Examples
4 Basic properties
- 4.1 Duality
- 4.2 Metric space structure
5 References

[edit] Motivation

A common problem is to find a good notion of a measure on a topological space that is compatible with the topology in some sense. One way to do this is to define a measure on the Borel sets of the topological space. In general there are several problems with this: for example, such a measure may not have a well defined support. Another approach to measure theory is to restrict to locally compact Hausdorff spaces, and only consider the measures that correspond to positive linear functionals on the space of continuous functions with compact support (some authors use this as the definition of a Radon measure). This produces a good theory with no pathological problems, but does not apply to spaces that are not locally compact.

The theory of Radon measures has most of the good properties of the usual theory for locally compact spaces, but applies to all Hausdorff topological spaces. The idea of the definition of a Radon measure is to find some properties that characterize the measures on locally compact spaces corresponding to positive functionals, and use these properties as the definition of a Radon measure on an arbitrary Hausdorff space.

[edit] Definitions

We let m be a measure on the σ-algebra of Borel sets of a Hausdorff topological space X.

The measure m is called inner regular or tight if m(B) is the supremum of m(K) for K a compact set contained in the Borel set B.

The measure m is called outer regular if m(B) is the infimum of m(U) for U an open set containing the Borel set B.

The measure m is called locally finite if every point has a neighborhood of finite measure.

The measure m is called a Radon measure if it is inner regular and locally finite.

(It is possible to extend the theory of Radon measures to non-Hausdorff spaces, essentially by replacing the word "compact" by "closed compact" everywhere. However there seem to be almost no applications of this extension.)

[edit] Examples

The following are all examples of Radon measures:

Dirac measure on any toplogical space;
Gaussian measure on Euclidean space $\mathbb{R}^{n}$ with its Borel topology and sigma algebra;
Lebesgue measure on Euclidean space.

Counting measure on Euclidean space is an example of a measure that is not a Radon measure, since it is not locally finite.

[edit] Basic properties

[edit] Duality

On a locally compact Hausdorff space, Radon measures correspond to positive linear functionals on the space of continuous functions with compact support. This is not surprising as this property is the main motivation for the definition of Radon measure.

[edit] Metric space structure

The space $\mathcal{M}_{+} (X)$ of all (positive) Radon measures on $X$ can be given the structure of a complete metric space by defining the Radon distance between two measures $m_{1}, m_{2} \in \mathcal{M}_{+} (X)$ to be

$\rho (m_{1}, m_{2}) := \sup \left\{ \left. \int_{X} f(x) \, \mathrm{d} (m_{1} - m_{2}) (x) \right| \mathrm{continuous\,} f : X \to [-1, 1] \subset \mathbb{R} \right\}.$

This metric has some limitations. For example, the space of Radon probability measures on $X$ ,

$\mathcal{P} (X) := \{ m \in \mathcal{M}_{+} (X) | m (X) = 1 \},$

is not sequentially compact with respect to the Radon metric: i.e., it is not guaranteed that any sequence of probability measures will have a subsequence that is convergent with respect to the Radon metric, which presents difficulties in certain applications. The Wasserstein metric is needed in order to make $\mathcal{P} (X)$ into a compact space.

Convergence in the Radon metric implies weak convergence of measures:

$\rho (m_{n}, m) \to 0 \implies m_{n} \rightharpoonup m,$

but the converse implication is false in general. Convergence of measures in the Radon metric is sometimes known as strong convergence, as contrasted with weak convergence.