Generalized N-dimensional Riemann Integral

Let $I=[a_{1},b_{1}]\times\cdots\times[a_{N},b_{N}]\subset\mathbb{R}^{N}$ be a compact interval, and let $f:I\to\mathbb{R}^{M}$ be a function. Let $\epsilon>0$ . If there exists a $y\in\mathbb{R}^{M}$ and a partition $P_{\epsilon}$ of $I$ such that for each refinement $P$ of $P_{\epsilon}$ (and corresponding Riemann Sum $S(f,P)$ ),

\left\|S(f,P)-y\right\|<\epsilon

Then we say that $f$ is Riemann integrable over $I$ , that $y$ is the Riemann integral of $f$ over $I$ , and we write

\int_{I}f:=\int_{I}f\,d\mu:=y

Note also that it is possible to extend this definition to more arbitrary sets; for any bounded set $D$ , one can find a compact interval $I$ such that $D\subset I$ , and define a function

\tilde{f}:I\to\mathbb{R}^{M}\quad x\mapsto\begin{cases}f(x),&x\in D\\ 0,&x\notin D\end{cases}

in which case we define

\int_{D}f:=\int_{I}\tilde{f}

Title	Generalized N-dimensional Riemann Integral
Canonical name	GeneralizedNdimensionalRiemannIntegral
Date of creation	2013-03-22 13:37:43
Last modified on	2013-03-22 13:37:43
Owner	vernondalhart (2191)
Last modified by	vernondalhart (2191)
Numerical id	6
Author	vernondalhart (2191)
Entry type	Definition
Classification	msc 26B12