Radon measure

Last updated

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. [1] 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.



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.


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 any open set U, m(U) is the supremum of m(K) over all compact subsets K of U.

The measure m is called outer regular if, for any Borel set B, m(B) is 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 (2004) harvtxt error: multiple targets (2×): CITEREFBourbaki2004 (help) 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 locally convex topology induced by the spaces ; this topology is finer than the topology of uniform convergence.

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,

Conversely, by the Riesz–Markov–Kakutani representation theorem, each positive linear form on 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 ; 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 Bourbaki (2004) harvtxt error: multiple targets (2×): CITEREFBourbaki2004 (help), Hewitt & Stromberg (1965) or Dieudonné (1970). 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 integralμ*(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.


The following are all examples of Radon measures:

The following are not examples of Radon measures:

We note that, intuitively, the Radon measure is useful in mathematical finance particularly for working with Lévy processes because it has the properties of both Lebesgue and Dirac measures, as unlike the Lebesgue, a Radon measure on a single point is not necessarily of measure 0. [2]

Basic properties

Moderated Radon measures

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

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 is given by Bourbaki (2004 , Exercise 5 of section 1) harvtxt error: multiple targets (2×): CITEREFBourbaki2004 (help) as follows. 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.

Radon spaces

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.

Metric space structure

The pointed cone of all (positive) Radon measures on can be given the structure of a complete metric space by defining the Radon distance between two measures to be

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

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 is a compact metric space, then the Wasserstein metric turns into a compact metric space.

Convergence in the Radon metric implies weak convergence of measures:

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.

Related Research Articles

In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets. Some authors require additional restrictions on the measure, as described below.

Compact space Topological notions of all points being "close"

In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

Measure (mathematics) Generalization of length, area, volume and integral

In mathematics, a measure on a set is a systematic way to assign a number to subsets of a set, intuitively interpreted as the size of the subset. Those sets which can be associated with such a number, we call measurable sets. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space. This assigns the usual length, area, or volume to certain subsets of the given Euclidean space. For instance, the Lebesgue measure of an interval of real numbers is its usual length, but also assigns numbers to other kinds of sets in a way that is consistent with the lengths of intervals.

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.

In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure.

Pontryagin duality Duality for locally compact abelian groups

In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group, the finite abelian groups, and the additive group of the integers, the real numbers, and every finite dimensional vector space over the reals or a p-adic field.

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In mathematics, a nuclear space is a topological vector space that can be viewed as a generalization of finite dimensional Euclidean spaces that is different from Hilbert spaces. Nuclear spaces have many of the desirable properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.

In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets.

In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets.

In mathematics, a positive measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number, and a set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is the countable union of measurable sets with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.e. all finite measures are σ-finite but there are (many) σ-finite measures that are not finite.

In measure theory, a branch of mathematics, a finite measure or totally finite measure is a special measure that always takes on finite values. Among finite measures are probability measures. The finite measures are often easier to handle than more general measures and show a variety of different properties depending on the sets they are defined on.

In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.

In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers. This space, denoted by , is a vector space with respect to the pointwise addition of functions and scalar multiplication by constants. It is, moreover, a normed space with norm defined by

In mathematics, lifting theory was first introduced by John von Neumann in a pioneering paper from 1931, in which he answered a question raised by Alfréd Haar. The theory was further developed by Dorothy Maharam (1958) and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea (1961). Lifting theory was motivated to a large extent by its striking applications. Its development up to 1969 was described in a monograph of the Ionescu Tulceas. Lifting theory continued to develop since then, yielding new results and applications.

Lebesgue integration

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.

In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named for Frigyes Riesz (1909) who introduced it for continuous functions on the unit interval, Andrey Markov (1938) who extended the result to some non-compact spaces, and Shizuo Kakutani (1941) who extended the result to compact Hausdorff spaces.


  1. Folland, Gerald (1999). Real Analysis: Modern techniques and their applications . New York: John Wiley & Sons, Inc. p.  212. ISBN   0-471-31716-0.
  2. Cont, Rama, and Peter Tankov. Financial modelling with jump processes. Chapman & Hall, 2004.
Functional-analytic development of the theory of Radon measure and integral on locally compact spaces.
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.
Contains a simplified version of Bourbaki's approach, specialised to measures defined on separable metrizable spaces .