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

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

If $f\in {ℛ}^{*}[a,b]$ 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.

Figure 1: Riemann sum over a $\delta$-fine partition