We are going to extend the concept of Riemann integral to functions of several variables.

Let $f\colon\mathbb R^n \to\mathbb R$ be a bounded function with compact support. Recalling the definitions of polyrectangle and the definitions of upper and lower Riemann sums on polyrectangles, we define $$ S^*(f) := \inf\{ S^*(f,P) \colon \text{$P$ is a polyrectangle, $f(x)=0$ for every $x\in\mathbb R^n\setminus \cup P$}\}, $$ $$ S_*(f) := \sup\{ S_*(f,P) \colon \text{$P$ is a polyrectangle, $f(x)=0$ for every $x\in\mathbb R^n\setminus \cup P$}\}. $$ If $S^*(f)=S_*(f)$ we say that $f$ is Riemann-integrable on $\mathbb R^n$ and we define the Riemann integral of $f$ : $$ \int f(x)\, dx := S^*(f) = S_*(f). $$

Clearly one has $S^*(f,P)\ge S_*(f,P)$ . Also one has $S^*(f,P)\ge S_*(f,P')$ when $P$ and $P'$ are any two polyrectangles containing the support of $f$ . In fact one can always find a common refinement $P''$ of both $P$ and $P'$ so that $S^*(f,P)\ge S^*(f,P'')\ge S_*(f,P'')\ge S_*(f,P')$ . So, to prove that a function is Riemann-integrable it is enough to prove that for every $\epsilon>0$ there exists a polyrectangle $P$ such that $S^*(f,P)-S_*(f,P)<\epsilon$ .

Next we are going to define the integral on more general domains. As a byproduct we also define the measure of sets in $\mathbb R^n$ .

Let $D\subset \mathbb R^n$ be a bounded set. We say that $D$ is Riemann measurable if the characteristic function

Let now $D\subset \mathbb R^n$ be a Riemann measurable set and let $f\colon D\to \mathbb R$ be a bounded function. We say that $f$ is Riemann measurable if the function $\bar f\colon\mathbb R^n\to\mathbb R$