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For example, sin(x) dx is a real-valued Radon measure, but is not even an extended signed measure as it cannot be written as the difference of two measures at least one of which is finite. Some authors use the preceding approach to define positive Radon measures to be the positive linear forms on K (X). [4]
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.
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 ...
The Wasserstein metric has a formal link with Procrustes analysis, with application to chirality measures, [6] and to shape analysis. [7] In computational biology, Wasserstein metric can be used to compare between persistence diagrams of cytometry datasets. [8] The Wasserstein metric also has been used in inverse problems in geophysics. [9]
is the pure point part (a discrete measure). The absolutely continuous measures are classified by the Radon–Nikodym theorem, and discrete measures are easily understood. Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures.
An important fact about the Bochner integral is that the Radon–Nikodym theorem fails to hold in general, and instead is a property (the Radon–Nikodym property) defining an important class of ″nice″ Banach spaces.
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 .
In measure theory, a radonifying function (ultimately named after Johann Radon) between measurable spaces is one that takes a cylinder set measure (CSM) on the first space to a true measure on the second space. It acquired its name because the pushforward measure on the second space was historically thought of as a Radon measure.