generalized Riemann integral
A gauge is a function which assigns to every real number an interval such that .
Given a gauge , a partition of an interval is said to be -fine if, for every point , the set containing is a subset of
A function is said to be generalized Riemann integrable on if there exists a number such that for every there exists a gauge on such that if is any -fine partition of , then
where is any Riemann sum for using the partition . The collection of all generalized Riemann integrable functions is usually denoted by .
If then the number is uniquely determined, and is called the generalized Riemann integral of over .
The reason that this is called a generalized Riemann integral is that, in the special case where for some number , we recover the Riemann integral as a special case.
Title | generalized Riemann integral |
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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 |