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The Lebesgue–Stieltjes integral ()is defined when : [,] is Borel-measurable and bounded and : [,] is of bounded variation in [a, b] and right-continuous, or when f is non-negative and g is monotone and right-continuous.
Henri Léon Lebesgue ForMemRS [1] (French: [ɑ̃ʁi leɔ̃ ləbɛɡ]; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of a function defined for that axis.
The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of ...
Lebesgue measure; Lebesgue integration; Lebesgue's density theorem; 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 ...
The Lebesgue integral, named after French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the Riemann integral , which it largely replaced in mathematical analysis since the first half of the 20th century.
The Laplace–Stieltjes transform of a real-valued function g is given by a Lebesgue–Stieltjes integral of the form ()for s a complex number.As with the usual Laplace transform, one gets a slightly different transform depending on the domain of integration, and for the integral to be defined, one also needs to require that g be of bounded variation on the region of integration.
In mathematics, the Kolmogorov integral (or Kolmogoroff integral) is a generalized integral introduced by Kolmogoroff including the Lebesgue–Stieltjes integral, the Burkill integral, and the Hellinger integral as special cases. The integral is a limit over a directed family of partitions, when the resulting limiting value is independent of ...
The Riemann–Stieltjes integral admits integration by parts in the form () = () () ()and the existence of either integral implies the existence of the other. [2]On the other hand, a classical result [3] shows that the integral is well-defined if f is α-Hölder continuous and g is β-Hölder continuous with α + β > 1 .