Search results
Results from the WOW.Com Content Network
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 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 .
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 ...
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.
The Radon–Nikodym theorem essentially states that, under certain conditions, any measure ν can be expressed in this way with respect to another measure μ on the same space. The function f is then called the Radon–Nikodym derivative and is denoted by d ν d μ {\displaystyle {\tfrac {d\nu }{d\mu }}} . [ 1 ]
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.