Generalised N-dimensional Riemann Sum

Let $I=[a_1,b_1]\times \mathrm{\cdots }\times [a_N,b_N]$ be an $N$-cell in ${ℝ}^N$. For each $j=1,\mathrm{\dots },N$, let be a partition $P_j$ of $[a_j,b_j]$. We define a partition $P$ of $I$ as

$$P:=P_1\times \mathrm{\cdots }\times P_N$$

Each partition $P$ of $I$ generates a subdivision of $I$ (denoted by ${(I_{\nu })}_{\nu }$) of the form

$$I_{\nu }=[t_{1,j},t_{1,j+1}]\times \mathrm{\cdots }\times [t_{N,k},t_{N,k+1}]$$

Let $f:U\to {ℝ}^M$ be such that $I\subset U$, and let ${(I_{\nu })}_{\nu }$ be the corresponding subdivision of a partition $P$ of $I$. For each $\nu$, choose $x_{\nu }\in I_{\nu }$. Define

$$S(f,P):=\sum _{\nu }f(x_{\nu })\mu (I\nu )$$

As the Riemann sum of $f$ corresponding to the partition $P$.

A partition $Q$ of $I$ is called a refinement of $P$ if $P\subset Q$.