# 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 $[a,b]\subset\mathbb{R}$. 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