|
A gauge  is a function which assigns to every real number  an interval
 such that
 .
Given a gauge , a partition
 of an interval ![$ [a,b]$](http://images.planetmath.org:8080/cache/objects/4328/l2h/img6.png "$ [a,b]$") is said to be -fine if, for every point
![$ x \in [a,b]$](http://images.planetmath.org:8080/cache/objects/4328/l2h/img7.png "$ x \in [a,b]$") , the set  containing is a subset of
A function
![$ f : [a, b] \rightarrow \mathbb{R}$](http://images.planetmath.org:8080/cache/objects/4328/l2h/img9.png "$ f : [a, b] \rightarrow \mathbb{R}$") is said to be generalized Riemann integrable on if there exists a number
 such that for every
 there exists a gauge
 on such that if
 is any
-fine partition of , then
where
 is any Riemann sum for  using the partition
. The collection of all generalized Riemann integrable functions is usually denoted by
![$ \mathcal{R}^{*}[a,b]$](http://images.planetmath.org:8080/cache/objects/4328/l2h/img17.png "$ \mathcal{R}^{*}[a,b]$") .
If
![$ f \in \mathcal{R}^{*}[a,b]$](http://images.planetmath.org:8080/cache/objects/4328/l2h/img18.png "$ f \in \mathcal{R}^{*}[a,b]$") then the number  is uniquely determined, and is called the generalized Riemann integral of over .
The reason that this is called a generalized Riemann integral is that, in the special case where
![$ \delta (x) = [x - y, x + y]$](http://images.planetmath.org:8080/cache/objects/4328/l2h/img20.png "$ \delta (x) = [x - y, x + y]$") for some number  , we recover the Riemann integral as a special case.
Figure: Riemann sum over a -fine partition
|
|
| 
