Radon space
Encyclopedia
In mathematics
, a Radon space, named after Johann Radon
, is a separable metric space
(M, d) such that every Borel probability measure
on M is inner regular
. Since a probability measure is globally finite, and hence a locally finite measure
, every probability measure on a Radon space is also a Radon measure
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Radon space, named after Johann Radon
Johann Radon
Johann Karl August Radon was an Austrian mathematician. His doctoral dissertation was on calculus of variations .- Life :...
, is a separable metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
(M, d) such that every Borel probability measure
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
on M is inner regular
Inner regular measure
In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.-Definition:...
. Since a probability measure is globally finite, and hence a locally finite measure
Locally finite measure
In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.-Definition:...
, every probability measure on a Radon space is also a Radon measure
Radon measure
In mathematics , 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 locally finite and inner regular.-Motivation:...
.