fundamental theorems of calculus for Lebesgue integration


Loosely, the Fundamental Theorems of CalculusMathworldPlanetmathPlanetmath serve to demonstrate that integration and differentiationMathworldPlanetmath are inversePlanetmathPlanetmathPlanetmath processes. Suppose that F(x) is an absolutely continuous function on an interval [a,b]. The two following forms of the theorem are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

First form of the Fundamental Theorem:

There exists a function f(t) Lebesgue-integrable on [a,b] such that for any x[a,b], we have F(x)-F(a)=axf(t)𝑑t.

Second form of the Fundamental Theorem:

F(x) is differentiableMathworldPlanetmathPlanetmath almost everywhere on [a,b] and its derivativePlanetmathPlanetmath, denoted F(x), is Lebesgue-integrable on that interval. In addition, we have the relationMathworldPlanetmath F(x)-F(a)=axF(t)𝑑t for any x[a,b].

Title fundamental theorems of calculus for Lebesgue integration
Canonical name FundamentalTheoremsOfCalculusForLebesgueIntegration
Date of creation 2013-03-22 12:27:54
Last modified on 2013-03-22 12:27:54
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 17
Author mathcam (2727)
Entry type Theorem
Classification msc 26-00
Synonym first fundamental theorem of calculus
Synonym second fundamental theorem of calculus
Synonym fundamental theorem of calculus
Related topic FundamentalTheoremOfCalculusClassicalVersion
Related topic FundamentalTheoremOfCalculusForRiemannIntegration
Related topic ChangeOfVariableInDefiniteIntegral