Darboux Sums in Rn

Def: Let $R$ be a rectangle in ${\mathbb{R}}^n$ if it is of the form

where each $[a_i,b_i]$ is a closed interval of real numbers. The number

will be called the diagonal of $R$, and the number

will be called the measure of $R$. In the case that $m(R)>0$, then $R$ is called non-degenerate.

Def: Let $R=\prod _{i=1}^n[a_i,b_i]$. If ${\mathcal{P}}_i$ is a partition of the interval $[a_i,b_i]$, for each $1\le i\le n$, we say that

is a partition of $R$. Note that $\mathcal{P}$ is a finite subset of $R$, that consists of the vertices of each of the subrectangles of $R$ induced by $\mathcal{P}$. The partition ${\mathcal{P}}_{\mathcal{i}}$ is called the $i$th coordinate partition of $\mathcal{P}$. We will denote ${\mathbf{P}}_R$ as the set of all partitions of the rectangle $R$.

Lemma: Let $R\subseteq {\mathbb{R}}^n$ be a rectangle for any $\epsilon >0$ there’s a $R\subseteq R^{\prime }$ with rational dimensions such that $m(R^{\prime })-m(R)<\epsilon$

Def: Let $\mathcal{P}$ and $\mathcal{Q}$ be two partitions of $R$, with $\mathcal{P}\mathcal{=}\prod _{\mathcal{i}\mathcal{=}\mathcal{1}}^{\mathcal{n}}{\mathcal{P}}_{\mathcal{i}}$ and $\mathcal{Q}\mathcal{=}\prod _{\mathcal{i}\mathcal{=}\mathcal{1}}^{\mathcal{n}}{\mathcal{Q}}_{\mathcal{i}}$. We say that $\mathcal{Q}$ **refines $\mathcal{P}$ if for all $1\le i\le n$, ${\mathcal{P}}_{\mathcal{i}}\subseteq {\mathcal{Q}}_{\mathcal{i}}$. This is equivalent to $\mathcal{P}\subseteq \mathcal{Q}$.

Def: Given two partitions of $R$, $\mathcal{P}$ and $\mathcal{Q}$ there’s not necessarily a relationship of refinement between each other, but there exists a third partition that refines both $\mathcal{P}$ and $\mathcal{Q}$ denoted as

Def: Let $f$ be a real-valued function defined and bounded on the rectangle $R\subseteq {\mathbb{R}}^n$ and $\mathcal{P}$ be a partition of $\mathcal{P}$. If $R_1,\dots ,R_k$ are the corresponding subrectangles of $R$ induced by $\mathcal{P}$, we can define the lower sum of $f$over the partition $\mathcal{P}$, denoted as $L(f,\mathcal{P}\mathcal{)}$ or $\underset{―}{S}(f,\mathcal{P}\mathcal{)}$

Similarly, we can define the ************upper sum of $f$ over the partition $\mathcal{P}$, denoted as $U(f,\mathcal{P}\mathcal{)}$ or $\bar{S}(f,P)$

Prop: Let $\mathcal{P}$ be any partition of $R$, then $L(f,\mathcal{P})\le U(f,\mathcal{P}\mathcal{)}$ .

Prop: Let $\mathcal{P}\mathcal{,}\mathcal{Q}\in {\mathbf{P}}_R$. If $\mathcal{Q}$ refines $\mathcal{P}$ then $L(f,\mathcal{P})\le L(f,\mathcal{Q}\mathcal{)}$ and $U(f,\mathcal{Q})\le U(f,\mathcal{P}\mathcal{)}$

Prop: Let $\mathcal{P}$ and $\mathcal{Q}$ be partitions of $R$, then $L(f,\mathcal{P})\le U(f,\mathcal{Q}\mathcal{)}$

Def: Since the following sets are bounded $\{L(f,\mathcal{P})\mid \mathcal{P}\in {\mathbf{P}}_R\}$ and $\{U(f,\mathcal{P})\mid \mathcal{P}\in {\mathbf{P}}_R\}$, then we can consider their supremum and infimum respectively. Then

and

are called the **************lower integral $f$ over $R$ and ********the upper integral of $f$ over $R$. In general it is true that