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]⊂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∈[a,b], we have F(x)-F(a)=∫xaf(t)𝑑t.
Second form of the Fundamental Theorem:
F(x) is differentiable almost everywhere on [a,b] and its derivative
, denoted F′(x), is Lebesgue-integrable on that interval. In addition, we have the relation
F(x)-F(a)=∫xaF′(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 |