Radon measure

For more details, see the article on Radon measures.

Let be a Radon measure and some point in Euclidean space.

On a strongly Lindelöf space every Radon measure is moderated.

By definition, any Radon measure is locally finite.

These real-valued Radon measures need not be signed measures.

It acquired its name because the pushforward measure on the second space was historically thought of as a Radon measure.

Hence, δ is also a Radon measure.

This is not surprising as this property is the main motivation for the definition of Radon measure.

If μ is both inner regular and locally finite, it is called a Radon measure.

Radon measures on arbitrary topological spaces and cylindrical measures.

In measure theory, a field of mathematics, the Hausdorff density measures how concentrated a Radon measure is at some point.

Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures.

The functional I defines a Radon measure on [a,b].

In fact, it is the vertex of the pointed cone of all non-negative Radon measures on X.

One approach to measure theory is to start with a Radon measure, defined as a positive linear functional on C(X).

L. Schwartz, Radon measures.

Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support.

From the definition of the vector Radon measure and from the properties of the perimeter, the following formula holds true:

This gives an identification of real-valued Radon measures with the dual space of the locally convex space .

One can more generally develop a general theory of Radon measures as distributional sections of using the Riesz representation theorem.

It defines a Radon measure V on the Grassmanian bundle of ℝ'

By the weak compactness of Radon measures, Tan(μ, a) is nonempty if one of the following conditions hold:

The support (in the sense of distributions) of the (vector) Radon measure is a subset of the boundary of , .

Then no measure which vanishes at points on Z is a Radon measure, since any compact set in Z is countable.

This is due to , and is often used to define the integral with respect to vector measures , and especially vector-valued Radon measures.