Radon measure

In what follows X denotes a locally compact topological space. The continuous real-valued functions with compact support on X form a vector space K(X) = C_C(X), which can be given a natural locally convex topology. Indeed, K(X) is the union of the spaces K(X, K) of continuous functions with support contained in compact sets K. Each of the spaces K(X, K) carries naturally the topology of uniform convergence, which makes it into a Banach space. But as a union of topological spaces is a special case of a direct limit of topological spaces, the space K(X) can be equipped with the direct limit locally convex topology induced by the spaces K(X, K); this topology is finer than the topology of uniform convergence.

If m is a Radon measure on ${\displaystyle X,}$ then the mapping

$${\displaystyle I:f\mapsto \int f(x)\,m(dx)}$$

is a continuous positive linear map from K(X) to R. Positivity means that I(f) ≥ 0 whenever f is a non-negative function. Continuity with respect to the direct limit topology defined above is equivalent to the following condition: for every compact subset K of X there exists a constant M_K such that, for every continuous real-valued function f on X with support contained in K,

$${\displaystyle |I(f)|\leq M_{K}\sup _{x\in X}|f(x)|.}$$

Conversely, by the Riesz–Markov–Kakutani representation theorem, each positive linear form on K(X) arises as integration with respect to a unique regular Borel measure.

A real-valued Radon measure is defined to be any continuous linear form on K(X); they are precisely the differences of two Radon measures. This gives an identification of real-valued Radon measures with the dual space of the locally convex space K(X). These real-valued Radon measures need not be signed measures. For example, sin(x) dx is a real-valued Radon measure, but is not even an extended signed measure as it cannot be written as the difference of two measures at least one of which is finite.

Some authors use the preceding approach to define (positive) Radon measures to be the positive linear forms on K(X).[3] In this set-up it is common to use a terminology in which Radon measures in the above sense are called positive measures and real-valued Radon measures as above are called (real) measures.

To complete the buildup of measure theory for locally compact spaces from the functional-analytic viewpoint, it is necessary to extend measure (integral) from compactly supported continuous functions. This can be done for real or complex-valued functions in several steps as follows:

1. Definition of the upper integral μ*(g) of a lower semicontinuous positive (real-valued) function g as the supremum (possibly infinite) of the positive numbers μ(h) for compactly supported continuous functions h ≤ g;
2. Definition of the upper integral μ*(f) for an arbitrary positive (real-valued) function f as the infimum of upper integrals μ*(g) for lower semi-continuous functions g ≥ f;
3. Definition of the vector space F = F(X, μ) as the space of all functions f on X for which the upper integral μ*(|f|) of the absolute value is finite; the upper integral of the absolute value defines a semi-norm on F, and F is a complete space with respect to the topology defined by the semi-norm;
4. Definition of the space L¹(X, μ) of integrable functions as the closure inside F of the space of continuous compactly supported functions.
5. Definition of the integral for functions in L¹(X, μ) as extension by continuity (after verifying that μ is continuous with respect to the topology of L¹(X, μ));
6. Definition of the measure of a set as the integral (when it exists) of the indicator function of the set.

It is possible to verify that these steps produce a theory identical with the one that starts from a Radon measure defined as a function that assigns a number to each Borel set of X.

The Lebesgue measure on R can be introduced by a few ways in this functional-analytic set-up. First, it is possibly to rely on an "elementary" integral such as the Daniell integral or the Riemann integral for integrals of continuous functions with compact support, as these are integrable for all the elementary definitions of integrals. The measure (in the sense defined above) defined by elementary integration is precisely the Lebesgue measure. Second, if one wants to avoid reliance on Riemann or Daniell integral or other similar theories, it is possible to develop first the general theory of Haar measures and define the Lebesgue measure as the Haar measure λ on R that satisfies the normalisation condition λ([0, 1]) = 1.

Given a Radon measure m on a space X, we can define another measure M (on the Borel sets) by putting

$${\displaystyle M(B)=\inf\{m(V)\mid V{\text{ is an open set with }}B\subseteq V\subseteq X\}.}$$

The measure M is outer regular, and locally finite, and inner regular for open sets. It coincides with m on compact and open sets, and m can be reconstructed from M as the unique inner regular measure that is the same as M on compact sets. The measure m is called moderated if M is σ-finite; in this case the measures m and M are the same. (If m is σ-finite this does not imply that M is σ-finite, so being moderated is stronger than being σ-finite.)

On a hereditarily Lindelöf space every Radon measure is moderated.

An example of a measure m that is σ-finite but not moderated as follows.[6] The topological space X has as underlying set the subset of the real plane given by the y-axis of points (0, y) together with the points (1/n, m/n²) with m, n positive integers. The topology is given as follows. The single points (1/n, m/n²) are all open sets. A base of neighborhoods of the point (0, y) is given by wedges consisting of all points in X of the form (u, v) with |v − y| ≤ |u| ≤ 1/n for a positive integer n. This space X is locally compact. The measure m is given by letting the y-axis have measure 0 and letting the point (1/n, m/n²) have measure 1/n³. This measure is inner regular and locally finite, but is not outer regular as any open set containing the y-axis has measure infinity. In particular the y-axis has m-measure 0 but M-measure infinity.

A topological space is called a Radon space if every finite Borel measure is a Radon measure, and strongly Radon if every locally finite Borel measure is a Radon measure. Any Suslin space is strongly Radon, and moreover every Radon measure is moderated.

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.

The pointed cone 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 m₁, m₂ ∈ M₊(X) to be

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

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

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

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. On the other hand, if X is a compact metric space, then the Wasserstein metric turns P(X) into a compact metric space.

Convergence in the Radon metric implies weak convergence of measures:

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

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.