# generalized Riemann integral

A *gauge* $\delta $ is a function which assigns to every real number $x$ an interval $\delta (x)$ such that $x\in \delta (x)$.

Given a gauge $\delta $, a partition^{} $U_{i}{}_{i=1}{}^{n}$ of an interval $[a,b]$ is said to be $\delta $-fine if, for every point $x\in [a,b]$, the set ${U}_{i}$ containing $x$ is a subset of $\delta (x)$

A function $f:[a,b]\to \mathbb{R}$ is said to be generalized Riemann integrable on $[a,b]$ if there exists a number $L\in \mathbb{R}$ such that for every $\u03f5>0$ there exists a gauge ${\delta}_{\u03f5}$ on $[a,b]$ such that if $\dot{\mathcal{P}}$ is any ${\delta}_{\u03f5}$-fine partition of $[a,b]$, then

$$ |

where $S(f;\dot{\mathcal{P}})$ is any Riemann sum^{} for $f$ using the partition $\dot{\mathcal{P}}$. The collection^{} of all generalized Riemann integrable functions is usually denoted by ${\mathcal{R}}^{*}[a,b]$.

If $f\in {\mathcal{R}}^{*}[a,b]$ then the number $L$ is uniquely determined, and is called the generalized Riemann integral of $f$ over $[a,b]$.

The reason that this is called a generalized Riemann integral is that, in the special case where $\delta (x)=[x-y,x+y]$ for some number $y$, we recover the Riemann integral as a special case.

Title | generalized Riemann integral |
---|---|

Canonical name | GeneralizedRiemannIntegral |

Date of creation | 2013-03-22 13:40:03 |

Last modified on | 2013-03-22 13:40:03 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 12 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 26A42 |

Synonym | Kurzweil-Henstock integral |

Synonym | gauge integral |

Defines | generalized Riemann integrable |

Defines | gauge |