Radon measure

The Radon measure is a measure on the sigma-algebra of Borel sets on a Hausdorff topological space X , which is locally finite and internal regular.

Content

Definition

Let μ be a measure on the sigma-algebra of Borel sets in a Hausdorff topological space X.

A measure μ is called internally regular if, for any Borel set B , μ ( B ) coincides with the supremum μ ( K ) for compact subsets of K in B.

A measure μ is called external regular if, for any Borel set B , μ ( B ) is the infimum of μ ( U ) over all open sets U containing B.

A measure μ is called locally finite if every point in X has a neighborhood U for which the value μ ( U ) is finite. (If μ is locally finite, then μ is finite on compact sets.)

A measure μ is called a Radon measure if it is internally regular and locally finite.

Note

The definition can be generalized to non-Hausdorff spaces, replacing the words “compact” with “closed and compact” everywhere, but this generalization has no applications yet.

Examples

Examples of Radon measures:

Lebesgue measure on Euclidean space (bounded on Borel subsets);
Haar measure on any locally compact topological group;
Dirac measure on any topological space;
Gaussian measures in Euclidean space $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ with his Borel sigma-algebra;
Probabilistic measures on the σ-algebra of Borel sets of any Polish space. This example not only generalizes the previous example, but includes many measures on locally compact spaces, for example, the Wiener measure on the space of real continuous functions on the interval [0,1].

The following measures are not Radon measures:

The counting measure on Euclidean space is not a Radon measure, since it is not locally finite.
The space of ordinals up to the first uncountable ordinal with order topology is a compact topological space. A measure that is equal to 1 on any set containing an uncountable closed set, and 0 otherwise, is a Borel measure, but it is not a Radon measure.
Let X be the set [0,1) equipped with the arrow topology . The Lebesgue measure on this topological space is not a Radon measure, since it is not internally regular. The latter follows from the fact that in this topology compact sets are no more than countable.
Standard measure of product on $(0,1)^{\kappa }$ ${\ displaystyle (0,1) ^ {\ kappa}}$ with uncountable $\kappa$ ${\ displaystyle \ kappa}$ - not a Radon measure, since any compact set is contained within the product of an uncountable number of closed intervals, the measure of each of which is less than 1.

Properties

Further, X denotes a locally compact topological space , μ is the Radon measure on $X$ ${\ displaystyle X}$ .

The measure μ defines a linear functional on the space of all compactly supported functions on X , that is, continuous functions with compact support:
$I\colon f\mapsto \int \limits _{X}f\,d\mu$ ${\ displaystyle I \ colon f \ mapsto \ int \ limits _ {X} f \, d \ mu}$

Moreover:

This functionality fully determines the measure itself.
This functionality is continuous and positive. Positivity means that $I(f)\geqslant 0$ ${\ displaystyle I (f) \ geqslant 0}$ , if a $f\geqslant 0$ ${\ displaystyle f \ geqslant 0}$ .

Radon Metric

The cone of all Radon measures on $X$ ${\ displaystyle X}$ you can give the structure of a complete metric space . Distance between two Radon measures $\mu _{1},\mu _{2}$ ${\ displaystyle \ mu _ {1}, \ mu _ {2}}$ is defined as follows:

\rho (\mu _{1},\mu _{2})=\sup \left\{\int \limits _{X}f(x)\,d(\mu _{1}-\mu _{2})(x)\right\},

{\ displaystyle \ rho (\ mu _ {1}, \ mu _ {2}) = \ sup \ left \ {\ int \ limits _ {X} f (x) \, d (\ mu _ {1} - \ mu _ {2}) (x) \ right \},}

where the supremum is taken over all continuous functions $f\colon X\to [-1,1]$ ${\ displaystyle f \ colon X \ to [-1,1]}$

This metric is called the Radon metric . The convergence of measures in the Radon metric is sometimes called strong convergence .

The space of Radon probability measures on $X$ ${\ displaystyle X}$ ,

{\mathcal {P}}(X):=\{\mu \mid \mu (X)=1\},

{\ displaystyle {\ mathcal {P}} (X): = \ {\ mu \ mid \ mu (X) = 1 \},}

is not sequentially compact with respect to this metric, that is, it is not guaranteed that any sequence of probability measures will have a subsequence that converges.

Convergence in the Radon metric implies weak convergence of measures:

\rho (\mu _{n},\mu )\to 0\Rightarrow \mu _{n}\rightharpoonup \mu

{\ displaystyle \ rho (\ mu _ {n}, \ mu) \ to 0 \ Rightarrow \ mu _ {n} \ rightharpoonup \ mu}

The converse is not true in the general case.

Integration

The definition of the integral over a wider class of functions (with optionally with a compact carrier) is carried out in several steps:

The upper integral μ * (g) of lower semicontinuous positive (real) functions g is defined as the supremum (possibly infinite) of positive numbers μ ( h ) for compactly supported continuous functions h ≤ g .
The upper integral μ * ( f ) for an arbitrary positive real-valued function f is defined as the infimum of the upper integrals μ * (g) for lower semi-continuous functions g ≥ f .
The vector space F = F ( X ; μ ) is defined as the space of all functions f on X for which the upper integral μ * (| f |) is finite; the upper integral of the absolute value defines a seminorm on F , and F is a complete space with respect to the topology defined by this seminorm.
The space L ¹ ( X , μ ) of integrable functions is defined as the closure in F of the space of continuous compactly supported functions.
The integral for functions from L ¹ ( X , μ ) is determined through the extension by continuity (after checking that μ is continuous with respect to the topology of L ¹ ( X , μ )).
The measure of the set is defined as the integral (when it exists) of the function of the indicator of the set.

One can verify that these actions give a theory identical to that which begins with the Radon measure, defined as a function that assigns a number to each Borel set in X.

Literature

Bourbaki, Nicolas (2004), Integration I , Springer Verlag , ISBN 3-540-41129-1 .
Dieudonné, Jean (1970), Treatise on analysis , vol. 2, Academic Press
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
Schwartz, Laurent (1974), Radon measures on arbitrary topological spaces and cylindrical measures , Oxford University Press, ISBN 0-19-560516-0

Links

RA Minlos (2001), "Radon measure" , in Hazewinkel, Michiel, Encyclopedia of Mathematics , Springer , ISBN 978-1-55608-010-4