Is there a definition of Riemann integration in $\mathbb{R^n}$? I know that Riemann integration is well-defined for a function $f: \mathbb{R} \to \mathbb{R}$.
I would like to ask whether there is a definition of Riemann integration for a function $f: \mathbb{R^n} \to \mathbb{R}$ ? If there is not, are there technical reasons to prevent such a definition ?
The reason for the question is to understand more about the motivation to construct the Lebesgue integration.
Thanks for your explanation!
 A: There is such a notion, which is used often in multivariable calculus before one introduces the Lebesgue integral. The motivation is the same - we want the volume under the graph of $f: \mathbb R^n \longrightarrow \mathbb R$. The approach of summing the area of rectangles with Riemann sums generalizes to summing the volume of higher dimensional rectangles.
Formally, a single variable Riemann sum of a bounded function $f: [a, b] \longrightarrow \mathbb R$ is a sum of the form $\sum_{i=0}^n f(t_i^*) (t_{i + 1} - t_i)$ where $a = t_0 < \dots < \ t_n = b$ is a partition of $[a, b]$ and $t_i^* \in [t_i, t_{i+1}]$. That is, we sample $f$ on these subintervals and sum the corresponding rectangle's areas. We say that $f$ is Riemann integrable if the limit of these expressions as the mesh size $\delta = \sup (t_{i+1} - t_i)$ goes to $0$. This limit is called the Riemann integral of $f$ and is denoted $\int_a^b f(x) dx$.
With this review aside, we can see a clear path to generalizing this to multivariable calculus. Take a rectanglesm $R = [a_1, b_1] \times \dots \times [a_n, b_n]$ and let $f: R \longrightarrow \mathbb R$ be bounded. Previously, we took smaller and smaller partitions of our interval $[a, b]$ into subintervals. Here, we'll partition our rectangle $R$ into subrectangles. Specifically, we consider partitions $R = \bigcup R_i$ where $R_i$ is a product of subintervals of $[a_i, b_i]$. We consider only "almost disjoint" unions $R_i$, meaning that if $i \neq j$ then $R_i$ and $R_j$ can intersect only on their boundaries. You can picture a partition of, say, a rectangle in $\mathbb R^2$ by drawing finitely many lines parallel to the axes which start from one edge and go to the parallel edge. The resulting "brick wall" is a partition. For higher dimensions, you do this with hyperplanes instead of lines.
Now, given such a partition, a Riemann sum for this partition is a sum of the form $\sum_i f(R_i^*) m(R_i)$ where $R_i^* \in R_i$ and $m(R_i)$ is the volume of the rectangle $R_i$, which is defined as the product of its side lengths. This is just like the single variable case - we sample $f$ along a partition and approximate the volume under its graph by these Riemann sums.
To finally define an integral, we need a limit of these sums. We define the diamter of a rectangle to be the largest distance between any two points in the rectangle (which is realized as the long diagonal). The mesh size of a partition $\bigcup R_i$ is the largest diameter of the $R_i$. Then we say $f$ is Riemann integrable if the limit of these Riemann sums as the mesh size goes to $0$ exists, and this limit is called the Riemann integral of $f$ and is denoted $\int_R f(x_1, \dots, x_n) dx_1 \dots dx_n$.
