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.
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.)
When the underlying measure space is a locally compact topological space, the definition of a Radon measure can be expressed in terms of continuous linear functionals on the space of continuous functions with compact support. This makes it possible to develop measure and integration in terms of functional analysis, an approach taken by and a number of other authors.
In what follows X denotes a locally compact topological space. The continuous real-valued functions with compact support on X form a vector space , which can be given a natural locally convex topology. Indeed, is the union of the spaces of continuous functions with support contained in compact sets K. Each of the spaces 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 can be equipped with the direct limit topology induced by the spaces .
If m is a Radon measure on then the mapping
is a continuous positive linear map from 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 MK such that, for every continuous real-valued function f on X with support contained in K,
A real-valued Radon measure is defined to be any continuous linear form on ; 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 . 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 ; see , or . 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:
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.
The following are all examples of Radon measures:
The following are not examples of Radon measures:
On a strongly Lindelof space 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.
This metric has some limitations. For example, the space of Radon probability measures on ,
Convergence in the Radon metric implies weak convergence of measures: