Generalized N-dimensional Riemann Integral

Let $I=[a_1,b_1]\times \mathrm{\cdots }\times [a_N,b_N]\subset {ℝ}^N$ be a compact interval, and let $f:I\to {ℝ}^M$ be a function. Let $ϵ>0$. If there exists a $y\in {ℝ}^M$ and a partition $P_{ϵ}$ of $I$ such that for each refinement $P$ of $P_{ϵ}$ (and corresponding Riemann Sum $S(f,P)$),

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

$${\int }_{I}f:={\int }_{I}f𝑑\mu :=y$$

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

in which case we define

$${\int }_{D}f:={\int }_I\tilde{f}$$