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

 $|S(f;\dot{\mathcal{P}})-L|<\epsilon,$

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.