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A measure on ℝ is a Radon measure if and only if it is a locally finite Borel measure. [5] The following are not examples of Radon measures: Counting measure on Euclidean space is an example of a measure that is not a Radon measure, since it is not locally finite.
A measure is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider probability space.
The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind. Lebesgue–Stieltjes integrals , named for Henri Leon Lebesgue and Thomas Joannes Stieltjes , are also known as Lebesgue–Radon integrals or just Radon integrals , after Johann Radon , to whom much of the ...
Lebesgue measure is both locally finite and inner regular, and so it is a Radon measure. Lebesgue measure is strictly positive on non-empty open sets, and so its support is the whole of R n. If A is a Lebesgue-measurable set with λ(A) = 0 (a null set), then every subset of A is also a null set. A fortiori, every subset of A is measurable.
Occurrence example 1 ~0.027 Radon concentration at the shores of large oceans is typically 1 Bq/m 3. Radon trace concentration above oceans or in Antarctica can be lower than 0.1 Bq/m 3, [100] with changes in radon levels being used to track foreign pollutants. [101] 10: 0.27 Mean continental concentration in the open air: 10 to 30 Bq/m 3.
Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb measure. In physics an example of a measure is spatial distribution of mass (see for example, gravity potential), or another non-negative extensive property, conserved ...
A Radon space, named after Johann Radon, is a topological space on which 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. In particular a separable complete metric space (M, d) is a Radon ...
A similar argument to the Dirac measure example shows that = [,]. Note that the boundary points 0 and 1 lie in the support: any open set containing 0 (or 1) contains an open interval about 0 (or 1), which must intersect ( 0 , 1 ) , {\displaystyle (0,1),} and so must have positive μ {\displaystyle \mu } -measure.