fundamental theorems of calculus for Lebesgue integration

Loosely, the Fundamental Theorems of Calculus serve to demonstrate that integration and differentiation are inverse processes. Suppose that $F(x)$ is an absolutely continuous function on an interval $[a,b]\subset ℝ$. The two following forms of the theorem are equivalent.

First form of the Fundamental Theorem:

There exists a function $f(t)$ Lebesgue-integrable on $[a,b]$ such that for any $x\in [a,b]$, we have $F(x)-F(a)={\int }_a^{x}f(t)𝑑t$.

Second form of the Fundamental Theorem:

$F(x)$ is differentiable almost everywhere on $[a,b]$ and its derivative, denoted $F^{\prime }(x)$, is Lebesgue-integrable on that interval. In addition, we have the relation $F(x)-F(a)={\int }_a^x{F}^{\prime }(t)𝑑t$ for any $x\in [a,b]$.

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