# 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 GeneralizedNdimensionalRiemannIntegral 2013-03-22 13:37:43 2013-03-22 13:37:43 vernondalhart (2191) vernondalhart (2191) 6 vernondalhart (2191) Definition msc 26B12