In mathematics (specifically in measure theory), 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 finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets.[1] These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.
In mathematics (specifically in measure theory), a Radonmeasure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff...
his part in the Radon–Nikodym theorem; the Radonmeasure concept of measure as linear functional; the Radon transform, in integral geometry, based on integration...
Radon is a chemical element; it has symbol Rn and atomic number 86. It is a radioactive noble gas and is colorless and odorless. Of the three naturally...
λ(G \ A) = λ(A \ F) = 0. Lebesgue measure is both locally finite and inner regular, and so it is a Radonmeasure. Lebesgue measure is strictly positive on non-empty...
probability measure is globally finite, and hence a locally finite measure, every probability measure on a Radon space is also a Radonmeasure. In particular...
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)...
constant, a locally integrable function or, in more general settings, a Radonmeasure. In the first case, the constant, known as the rate or intensity, is...
probability measure that is neither inner regular nor outer regular. Borel regular measureRadonmeasure Regularity theorem for Lebesgue measure Billingsley...
Radon mitigation is any process used to reduce radon gas concentrations in the breathing zones of occupied buildings, or radon from water supplies. Radon...
natural topology, and a (Radon) measure is defined as a continuous linear functional on this space. The value of a measure at a compactly supported function...
compact closure, so is not an outer measure.) Cartan introduced another way of constructing Haar measure as a Radonmeasure (a positive linear functional on...
the total variation metric coincides with the Radon metric. If μ and ν are both probability measures, then the total variation distance is also given...
condition to be an inner regular measure, since singleton sets such as {x} are always compact. Hence, δx is also a Radonmeasure. Assuming that the topology...
general version in measure theory is the following: Theorem — Let X be a locally compact Hausdorff space equipped with a finite Radonmeasure μ, and let Y be...
both inner regular, outer regular, and locally finite, it is called a Radonmeasure. The real line R {\displaystyle \mathbb {R} } with its usual topology...
necessarily a Radonmeasure). Lebesgue measure is an example of a positive Radonmeasure. One particularly important class of Radonmeasures are those that...
continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative...
^{n}(K)\mid K\subseteq A,K{\text{ is compact}}\},} so Gaussian measure is a Radonmeasure; is not translation-invariant, but does satisfy the relation d...
some Radonmeasure. Generally, when the term Dirac delta function is used, it is in the sense of distributions rather than measures, the Dirac measure being...
The health effects of radon are harmful, and include an increased chance of lung cancer. Radon is a radioactive, colorless, odorless, tasteless noble gas...
{\displaystyle B} has the Radon–Nikodym property if B {\displaystyle B} has the Radon–Nikodym property with respect to every finite measure. Equivalent formulations...
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