In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets.
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.
Let m be a measure on the σ-algebra of Borel sets of a Hausdorff topological space X.
(It is possible to extend the theory of Radon measures to non-Hausdorff spaces, essentially by replacing the word "compact" by "closed compact" everywhere.
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 Bourbaki and a number of other authors.
[2] 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) = Cc(X), which can be given a natural locally convex topology.
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,
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.
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.
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.
On a hereditarily Lindelöf space every Radon measure is moderated.
This measure is inner regular and locally finite, but is not outer regular as any open set containing the y-axis has measure infinity.
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
For example, the space of Radon probability measures on X,
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.