RADON MEASURE

In mathematics, a 'Radon measure', named after Johann Radon, on a Hausdorff topological space ''X'' is defined in measure theory to be a measure on the σ-algebra of Borel sets of ''X'' that is locally finite and inner regular.

Contents

Motivation

Definitions

Radon measures on locally compact spaces

Measures

Integration

Examples

Basic properties

Duality

Radon measures are Borel measures

Metric space structure

References

External links

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.
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

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.)

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) and a number of other authors.

Measures

In what follows ''X'' denotes a locally compact topological space. The continuous real-valued functions on ''X'' form a vector space $mathcal\{K\}(X)$, which can be given a natural locally convex topology. Indeed, $mathcal\{K\}(X)$ is the union of the spaces $mathcal\{K\}(X,K)$ of continuous functions with support contained in compact sets ''K''. Each of the spaces $mathcal\{K\}(X,K)$ 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 $mathcal\{K\}(X)$ can be equipped with the direct limit topology induced by the spaces $mathcal\{K\}(X,K)$.
If μ is a Radon measure on X, then the mapping
:: $f\; mapsto\; int\; f\; dmu$
is a ''continuous'' linear map from $mathcal\{K\}(X)$ to 'R'. 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 ''M''_''K'' such that, for every continuous real-valued function ''f'' on ''X'' with ''support contained in K'',
:: $|mu(f)|\; leq\; M\_K\; .\; sup\_\{xin\; X\}\; |f(x)|.$
Conversely, by a version of the Riesz representation theorem, each continuous linear functional on $mathcal\{K\}(X)$ arises as integration with respect to a Radon measure. This gives the identification of Radon measures with the dual space of the locally convex space $mathcal\{K\}(X)$.
Integration with respect to a positive measure (one that assigns a non-negative number to each measurable set) obviously gives rise to a ''positive'' linear functional on $mathcal\{K\}(X)$, i.e., one that assigns a non-negative value to a non-negative function. On the other hand, it is possible to show that every positive linear functional on $mathcal\{K\}(X)$ is continuous (and hence corresponds to a Radon measure), and that moreover each Radon measure is the difference of two positive measures.
The definitions above apply also to the case complex-valued functions with compact support and complex-valued continuous functionals, producing the theory of complex Radon measures. Similarly, the concepts can be expanded to more general topological vector spaces in place of 'R' and 'C'.

Integration

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:
# Definition of the 'upper integral' μ
★ (''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''
# Definition of the upper integral μ
★ (''f'') for a arbitrary positive (real-valued) function ''f'' as the infimum of upper integrals μ
★ (''g'') for lower semi-continuous functions ''g''≥''f''
# 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
# Definition of the space ''LL''¹(''X'',μ) of 'integrable functions' as the closure of the space of continuous compactly supported functions
# Definition of the 'integral' for functions in ''LL''¹(''X'',μ) as extension by continuity (after verifying that μ is continuous with respect to the topology of ''LL''¹(''X'',μ))
# 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.

Examples

The following are all examples of Radon measures:

★ Lebesgue measure on Euclidean space;

★ Haar measure on any locally compact topological group;

★ Dirac measure on any toplogical space;

★ Gaussian measure on Euclidean space $mathbb\{R\}^\{n\}$ with its Borel topology and sigma algebra;
Counting measure on Euclidean space is an example of a measure that is not a Radon measure, since it is not locally finite.

Basic properties

Duality

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.

Radon measures are Borel measures

A Radon measure is also a Borel measure, since the Radon measure of any compact set is finite.
'Proof:' Let $K$ be a compact subset of a Hausdorff space $X$ and let $mu$ be a Radon measure on $X$. Since $mu$ is locally finite there exist open neighbourhoods $U\_x$ with for all $x\; in\; K$. Since is an open cover of $K$ there exists a finite subset $mathcal\{I\}$ of $K$ such that . Then, .

Metric space structure

The pointed cone $mathcal\{M\}\_\{+\}\; (X)$ 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 $m\_\{1\},\; m\_\{2\}\; in\; mathcal\{M\}\_\{+\}\; (X)$ to be
:$$
ho (m_{1}, m_{2}) := sup left{ left. int_{X} f(x) , mathrm{d} (m_{1} - m_{2}) (x)
ight| mathrm{continuous,} f : X o [-1, 1] subset mathbb{R}
ight}.
This metric has some limitations. For example, the space of Radon probability measures on $X$,
:$mathcal\{P\}\; (X)\; :=\; \{\; m\; in\; mathcal\{M\}\_\{+\}\; (X)\; |\; m\; (X)\; =\; 1\; \},$
is not sequentially compact with respect to the Radon metric: i.e., it is not guaranteed that any sequence of probability measures will have a subsequence that is convergent with respect to the Radon metric, which presents difficulties in certain applications. The Wasserstein metric is needed in order to make $mathcal\{P\}\; (X)$ into a compact space.
Convergence in the Radon metric implies weak convergence of measures:
:$$
ho (m_{n}, m) o 0 implies m_{n}
ightharpoonup m,
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.

References

★ .

★ Radon measures on arbitrary topological spaces and cylindrical measures, , Laurent, Schwartz, Oxford University Press, ,

★ Measure and integration: an advanced course in basic procedures and applications, , Heinz, König, New York: Springer, ,

External links

★