Multiple Riemann integral

A multiple Riemann integral of a bounded function $f:R\to\mathbb{R}$ over a rectangle $R=\prod_{i=1}^n [a_i,b_i]\subset \mathbb{R}^n$ is a number $I\in\mathbb{R}$ such that for every $\varepsilon>0$ there exists $\delta>0$ with the property that for every rectangular partition of $R$ with mesh $<\delta$ and every choice of tags (sample points) $\xi_j$ in each subrectangle $R_j$, the corresponding Riemann sum

satisfies $|S(f)-I|<\varepsilon$. In this case one writes $\int_R f = I$.

This extends the one-variable Riemann integral by using partitions built from products of partitions of intervals (a product rectangle is a Cartesian product of intervals). The link with iterated integrals is made precise by the Riemann–Fubini theorem under appropriate hypotheses.

Examples:

• If $f(x)\equiv c$ on $R$, then $\int_R f = c\,\operatorname{vol}(R)$.
• On $R=[-1,1]^n$, the function $f(x)=\mathbf{1}_{\{x_1>0\}}$ is Riemann integrable and $\int_R f = 2^{n-1}$ (the discontinuities lie on the hyperplane $x_1=0$).