# Riemann integral

Let $I=[a,b]$ be an interval of $\mathbb{R}$ and let $f\colon I\to\mathbb{R}$ be a bounded function. For any finite set of points $\{x_{0},x_{1},x_{2},\ldots,x_{n}\}$ such that $a=x_{0}, there is a corresponding partition $P=\{[x_{0},x_{1}),[x_{1},x_{2}),\ldots,[x_{n-1},x_{n}]\}$ of $I$.

Let $C(\epsilon)$ be the set of all partitions of $I$ with $\max(x_{i+1}-x_{i})<\epsilon$. Then let $S^{*}(\epsilon)$ be the infimum of the set of upper Riemann sums with each partition in $C(\epsilon)$, and let $S_{*}(\epsilon)$ be the supremum of the set of lower Riemann sums with each partition in $C(\epsilon)$. If $\epsilon_{1}<\epsilon_{2}$, then $C(\epsilon_{1})\subset C(\epsilon_{2})$, so $S^{*}(\epsilon)$ is decreasing (http://planetmath.org/IncreasingdecreasingmonotoneFunction) and $S_{*}(\epsilon)$ is increasing (http://planetmath.org/IncreasingdecreasingmonotoneFunction). Moreover, $\lvert S^{*}(\epsilon)\rvert$ and $\lvert S_{*}(\epsilon)\rvert$ are bounded by $(b-a)\sup_{x}\lvert f(x)\rvert$. Therefore, the limits $S^{*}=\lim_{\epsilon\to 0}S^{*}(\epsilon)$ and $S_{*}=\lim_{\epsilon\to 0}S_{*}(\epsilon)$ exist and are finite. If $S^{*}=S_{*}$, then $f$ is Riemann-integrable over $I$, and the Riemann integral of $f$ over $I$ is defined by

 $\int_{a}^{b}f(x)dx=S^{*}=S_{*}.$
Title Riemann integral RiemannIntegral 2013-03-22 11:49:24 2013-03-22 11:49:24 bbukh (348) bbukh (348) 14 bbukh (348) Definition msc 28-00 msc 26A42 RiemannSum Integral2 Riemann integrable