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. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.
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. If there is no restriction to non-negative measures and complex measures are allowed, then Radon measures can be defined as the continuous dual space on the space of continuous functions with compact support. If such a Radon measure is real then it can be decomposed into the difference of two positive measures. Furthermore, an arbitrary Radon measure can be decomposed into four positive Radon measures, where the real and imaginary parts of the functional are each the differences of two positive Radon measures. 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.
Definitions 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, for every open set U, m(U) equals the supremum of m(K) over all compact subsets K of U. The measure m is called outer regular if, for every Borel set B, m(B) equals the infimum of m(U) over all open sets U containing B. The measure m is called locally finite if every point of X has a neighborhood U for which m(U) is finite. If m is locally finite, then it follows that m is finite on compact sets, and for locally compact Hausdorff spaces, the converse holds, too. Thus, in this case, local finiteness may be equivalently replaced by finiteness on compact subsets. The measure m is called a Radon measure if it is inner regular and locally finite. In many situations, such as finite measures on locally compact spaces, this also implies outer regularity (see also Radon spaces). (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.)
Radon measures on locally compact spaces 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.
Measures 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. 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 X , {\displaystyle X,} then the mapping
I : f ↦ ∫ f ( x ) m ( d x ) {\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 MK such that, for every continuous real-valued function f on X with support contained in K,
| I ( f ) | ≤ M K sup x ∈ X | f ( x ) | . {\displaystyle |I(f)|\leq M_{K}\sup _{x\in X}|f(x)|.}
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