# 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 $F(x)-F(a)={\int}_{a}^{x}f(t)\mathit{d}t$.

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^{} $F(x)-F(a)={\int}_{a}^{x}{F}^{\prime}(t)\mathit{d}t$ 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 |