Inmathematics (specifically inmeasure theory), aRadon measure, named afterJohann Radon, is ameasure on theσ-algebra ofBorel sets of aHausdorff topological spaceX that is finite on allcompact sets,outer regular on all Borel sets, andinner regular onopen sets.^[1] These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used inmathematical analysis and innumber theory are indeed Radon measures.

A common problem is to find a good notion of a measure on atopological space that is compatible with the topology in some sense. One way to do this is to define a measure on theBorel sets of the topological space. In general there are several problems with this: for example, such a measure may not have a well definedsupport. Another approach to measure theory is to restrict tolocally compactHausdorff spaces, and only consider the measures that correspond to positivelinear functionals on the space ofcontinuous 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 ofcontinuous 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.

Letm be a measure on theσ-algebra ofBorel sets of a Hausdorff topological spaceX.

• The measurem is calledinner regular ortight if, for every open setU,m(U) equals thesupremum ofm(K) over all compact subsetsK ofU.
• The measurem is calledouter regular if, for every Borel setB,m(B) equals theinfimum ofm(U) over all open setsU containingB.
• The measurem is calledlocally finite if every point ofX has a neighborhoodU for whichm(U) is finite.

Ifm is locally finite, then it follows thatm 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 measurem is called aRadon 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 alsoRadon 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.)

When the underlying measure space is alocally compact topological space, the definition of a Radon measure can be expressed in terms ofcontinuouslinear functionals on the space ofcontinuous functions withcompact support. This makes it possible to develop measure and integration in terms offunctional analysis, an approach taken by Bourbaki and a number of other authors.^[2]

In what followsX denotes a locally compact topological space. The continuousreal-valued functions withcompact support onX form avector spaceK(X) =C_c(X), which can be given a naturallocally convex topology. Indeed,K(X) is the union of the spacesK(X,K) of continuous functions with support contained incompact setsK. Each of the spacesK(X,K) carries naturally the topology ofuniform convergence, which makes it into aBanach space. But as a union of topological spaces is a special case of adirect limit of topological spaces, the spaceK(X) can be equipped with the direct limitlocally convex topology induced by the spacesK(X,K); this topology is finer than the topology of uniform convergence.

Ifm is a Radon measure on${\displaystyle X,}$ then the mapping^[3]$${\displaystyle I:f\mapsto \int f(x)m(dx)}$$

is acontinuous positive linear map fromK(X) toR. Positivity means thatI(f) ≥ 0 wheneverf is a non-negative function. Continuity with respect to the direct limit topology defined above is equivalent to the following condition: for every compact subsetK ofX there exists a constantM_K such that, for every continuous real-valued functionf onX withsupport contained inK,$${\displaystyle |I(f)|\le M_K\underset{x\in X}{sup}|f(x)|.}$$

Conversely, by theRiesz–Markov–Kakutani representation theorem, eachpositive linear form onK(X) arises as integration with respect to a unique regular Borel measure.

Areal-valued Radon measure is defined to beany continuous linear form onK(X); they are precisely the differences of two Radon measures. This gives an identification of real-valued Radon measures with thedual space of thelocally convex spaceK(X). These real-valued Radon measures need not besigned 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 definepositive Radon measures to be the positive linear forms onK(X).^[4] In this set-up it is common to use a terminology in which Radon measures in the above sense are calledpositive 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 theupper integralμ*(g) of alower semicontinuous positive (real-valued) functiong as thesupremum (possibly infinite) of the positive numbersμ(h) for compactly supported continuous functionsh ≤g;
2. Definition of the upper integralμ*(f) for an arbitrary positive (real-valued) functionf as the infimum of upper integralsμ*(g) for lower semi-continuous functionsg ≥f;
3. Definition of the vector spaceF =F(X,μ) as the space of all functionsf onX for which the upper integralμ*(|f|) of the absolute value is finite; the upper integral of the absolute value defines asemi-norm onF, andF is acomplete space with respect to the topology defined by the semi-norm;
4. Definition of the spaceL¹(X,μ) ofintegrable functions as theclosure insideF of the space of continuous compactly supported functions.
5. Definition of theintegral for functions inL¹(X,μ) as extension by continuity (after verifying thatμ is continuous with respect to the topology ofL¹(X,μ));
6. Definition of the measure of a set as the integral (when it exists) of theindicator 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 eachBorel set ofX.

TheLebesgue measure onR 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 theDaniell integral or theRiemann 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 ofHaar measures and define the Lebesgue measure as the Haar measureλ onR that satisfies the normalisation conditionλ([0, 1]) = 1.

The following are all examples of Radon measures:

• Lebesgue measure on Euclidean space (restricted to the Borel subsets);
• Haar measure on anylocally compact topological group;
• Dirac measure on any topological space;
• Gaussian measure onEuclidean spaceℝⁿ with its Borel sigma algebra;
• Probability measures on theσ-algebra ofBorel sets of anyPolish space. This example not only generalizes the previous example, but includes many measures on non-locally compact spaces, such asWiener measure on the space of real-valued continuous functions on the interval[0, 1].
• Counting measure on any finite space
• A measure on${\displaystyle {\mathbb{R}}^n}$ is a Radon measure if and only if it is alocally finiteBorel measure.^[5][6]

The following are not examples of Radon measures:

• Counting measure on Euclidean space is an example of a measure that is not a Radon measure, since it is not locally finite.
• The space ofordinals at most equal toΩ, thefirst uncountable ordinal with theorder topology is a compact topological space. The measure which equals1 on any Borel set that contains an uncountable closed subset of[1, Ω), and0 otherwise, is Borel but not Radon, as the one-point set{Ω} has measure zero but any open neighbourhood of it has measure1.^[7]
• LetX be the interval[0, 1) equipped with the topology generated by the collection of half open intervals{[a,b) : 0 ≤a <b ≤ 1}. This topology is sometimes calledSorgenfrey line. On this topological space, standard Lebesgue measure is not Radon since it is not inner regular, since compact sets are at most countable.
• LetZ be aBernstein set in[0, 1] (or any Polish space). Then no measure which vanishes at points onZ is a Radon measure, since any compact set inZ is countable.
• Standardproduct measure on(0, 1)^κ for uncountableκ is not a Radon measure, since any compact set is contained within a product of uncountably many closed intervals, each of which is shorter than 1.

We note that, intuitively, the Radon measure is useful in mathematical finance particularly for working with Lévy processes because it has the properties of bothLebesgue andDirac measures, as unlike the Lebesgue, a Radon measure on a single point is not necessarily of measure0.^[8]

Given a Radon measurem on a spaceX, we can define another measureM (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 measureM is outer regular, and locally finite, and inner regular for open sets. It coincides withm on compact and open sets, andm can be reconstructed fromM as the unique inner regular measure that is the same asM on compact sets. The measurem is calledmoderated ifM isσ-finite; in this case the measuresm andM are the same. (Ifm isσ-finite this does not imply thatM isσ-finite, so being moderated is stronger than beingσ-finite.)

On ahereditarily Lindelöf space every Radon measure is moderated.

An example of a measurem that isσ-finite but not moderated as follows.^[9] The topological spaceX has as underlying set the subset of the real plane given by they-axis of points(0,y) together with the points(1/n,m/n²) withm,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 inX of the form(u,v) with|v −y| ≤ |u| ≤ 1/n for a positive integern. This spaceX is locally compact. The measurem is given by letting they-axis have measure0 and letting the point(1/n,m/n²) have measure1/n³. This measure is inner regular and locally finite, but is not outer regular as any open set containing they-axis has measure infinity. In particular they-axis hasm-measure0 butM-measure infinity.

A topological space is called aRadon space if every finite Borel measure is a Radon measure, andstrongly Radon if every locally finite Borel measure is a Radon measure. AnySuslin 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.

Thepointed coneM₊(X) of all (positive) Radon measures onX can be given the structure of acompletemetric space by defining theRadon distance between two measuresm₁,m₂ ∈M₊(X) to be$${\displaystyle \rho (m_1,m_2)=sup\left\{{\int }_{X}f(x)(m_1-m_2)(dx)|\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}f:X\to [-1,1]\subset \mathbb{R}\right\}.}$$

This metric has some limitations. For example, the space of Radonprobability measures onX,$${\displaystyle \mathcal{P}(X)=\{m\in {\mathcal{M}}_{+}(X)\mid m(X)=1\},}$$is notsequentially 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, ifX is a compact metric space, then theWasserstein metric turnsP(X) into a compact metric space.

Convergence in the Radon metric impliesweak 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 asstrong convergence, as contrasted with weak convergence.

• Radonifying function
• Vague topology

• Bogachev, Vladimir I. (2007), "Measures on topological spaces",Measure Theory, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 476–583,doi:10.1007/978-3-540-34514-5_7,ISBN 978-3-540-34513-8
• Bourbaki, Nicolas (2004a),Integration I,Springer Verlag,ISBN 3-540-41129-1. Functional-analytic development of the theory of Radon measure and integral on locally compact spaces.
• Bourbaki, Nicolas (2004b),Integration II,Springer Verlag,ISBN 3-540-20585-3. Haar measure; Radon measures on general Hausdorff spaces and equivalence between the definitions in terms of linear functionals and locally finite inner regular measures on the Borel sigma-algebra.
• Dieudonné, Jean (1970),Treatise on analysis, vol. 2, Academic Press. Contains a simplified version of Bourbaki's approach, specialised to measures defined on separable metrizable spaces.
• Folland, Gerald (1999),Real Analysis: Modern techniques and their applications, New York: John Wiley & Sons, Inc., p. 212,ISBN 0-471-31716-0
• Hewitt, Edwin; Stromberg, Karl (1965),Real and abstract analysis, Springer-Verlag
• König, Heinz (1997),Measure and integration: an advanced course in basic procedures and applications, New York: Springer,ISBN 3-540-61858-9{{citation}}: CS1 maint: publisher location (link)
• Schwartz, Laurent (1974),Radon measures on arbitrary topological spaces and cylindrical measures, Oxford University Press,ISBN 0-19-560516-0
• Teschl, Gerald,Topics in Real Analysis, (lecture notes)
• Treves, Francois (2006),Topological vector spaces, distributions and kernels, Mineola, N.Y: Dover Publications,ISBN 978-0-486-45352-1

• R. A. Minlos (2001) [1994],"Radon measure",Encyclopedia of Mathematics,EMS Press

• Measures (measure theory)
• Integral representations
• Lp spaces

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