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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.
For (,) a measurable space, a sequence μ n is said to converge setwise to a limit μ if = ()for every set .. Typical arrow notations are and .. For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ n of measures on the interval [−1, 1] given by μ n (dx) = (1 + sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A measure is a set function that assigns a consistent magnitude to the measurable subsets of a measurable 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 ...
It is known that in the dual space of any Montel space, a sequence converges in the strong dual topology if and only if it converges in the weak* topology, [25] which in particular, is the reason why a sequence of distributions converges (in the strong dual topology) if and only if it converges pointwise (this leads many authors to use ...
One application of this is to probability theory: for example, the central limit theorem is essentially a statement that if are the probability measures for certain sums of independent random variables, then converge weakly (and then vaguely) to a normal distribution, that is, the measure is "approximately normal" for large .
The Radon–Riesz property is a mathematical property for normed spaces that helps ensure convergence in norm. Given two assumptions (essentially weak convergence and continuity of norm), we would like to ensure convergence in the norm topology .
Specifically, if is a measure on (,), then has the Radon–Nikodym property with respect to if, for every countably-additive vector measure on (,) with values in which has bounded variation and is absolutely continuous with respect to , there is a -integrable function : such that = for every measurable set . [2]