GeneralizedRiemannIntegral 2003-06-08 07:43:05 2007-02-23 19:30:13 Definition generalized Riemann integrable gauge % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) \usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A \emph{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 \textbf{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 \textbf{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. \begin{figure}[!htb] \begin{center} \includegraphics{riemann.eps} \caption{Riemann sum over a $\delta$-fine partition} \end{center} \end{figure}