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Topic: Lebesgue integral

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Lebesgue integration

Dominated convergence theorem

Integral

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	Integral Article, Integral Information (Site not responding. Last check: 2007-10-21)
		Improper integrals usually turn up when the range of the function is infinite or, in the case of the Riemann integral, when the domain is infinite.
		The Lebesgue integral was created by Henri Lebesgue to integrate a wider class of functions and to prove very strong theorems about interchanging limits and integrals.
		the Riemann-Stieltjes integral, anextension of the Riemann integral
	www.anoca.org /integrals/integration/integral.html (1330 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		In the mathematical branch of real analysis, Lebesgue integration is a framework for extending the notion of integral as the area under the curve to a large class of functions whose domain may not even be in R.
		The Lebesgue approach is not the most elementary area-based integration theory; that distinction goes to the Riemann integral.
		Every reasonable notion of integral needs to be linear and monotone, and the Lebesgue integral is: if f and g are integrable functions and a and b are real numbers, then af + bg is integrable and ∫(af + bg) = a∫f + b∫g; if f ≤ g, then ∫f ≤ ∫g.
	www.informationgenius.com /encyclopedia/l/le/lebesgue_integration.html (2071 words)

	PlanetMath: Lebesgue integral (Site not responding. Last check: 2007-10-21)
		The Lebesgue integral equals the Riemann integral everywhere the latter is defined; the advantage to the Lebesgue integral is that it is often well defined even when the corresponding Riemann integral is undefined.
		For example, the Riemann integral of the characteristic function of the rationals in
		This is version 13 of Lebesgue integral, born on 2002-02-13, modified 2005-05-04.
	planetmath.org /encyclopedia/Integral2.html (153 words)

	Lebesgue integral (Site not responding. Last check: 2007-10-21)
		A Short Course on the Lebesgue Integral and...
		Relation of the Riemann integral to the Legesgue integral.
		Empowering Functions: A Case for the Lebesgue Integral...
	www.scienceoxygen.com /math/536.html (143 words)

	Distribution - Open Encyclopedia (Site not responding. Last check: 2007-10-21)
		Sometimes, people talk of "probability distribution" when they just mean "probability measure", especially if it is obtained by taking the product of the Lebesgue measure by a positive, real-valued measurable function of integral equal to 1.
		The function f : U → R is called locally integrable if it is Lebesgue integrable over every compact subset K of U.
		This extends the range of symbolic methods that can be made into rigorous mathematics, for example Feynman integrals.
	open-encyclopedia.com /Distribution (1635 words)

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