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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.
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 ...
Counting measure; Complete measure; Haar measure; Outer measure; Borel regular measure; Radon measure; Measurable function; Null set, negligible set; Almost everywhere, conull set; Lp space; Borel–Cantelli lemma; Lebesgue's monotone convergence theorem; Fatou's lemma; Absolutely continuous; Uniform absolute continuity; Total variation; Radon ...
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.
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 ...
Radon transform. Maps f on the (x, y)-domain to Rf on the (α, s)-domain.. In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line.
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 ...
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