For the multi-variable Riemann integral (specifically, integrating functions $f : \mathbb{R}^n \rightarrow \mathbb{R}$ over subsets of $\mathbb{R}^n$), the most common treatment (see footnote below) seems to be defining the integral in terms of rectangular partitions of the domain of integration (partition composed of n-dimensional "rectangles").
On the other hand, Courant & John's Introduction to Calculus and Analysis presents a more general definition (for the $\mathbb{R}^2$ case) involving partitions with fairly arbitrary "shape" subsets (not just rectangles):
Consider...Jordan-measurable set $R$ with area $A(R) = \Delta R$, and a function $f(x, y)$ that is continuous everywhere...We subdivide $R$ into $N$ nonoverlapping Jordan-measurable subsets...and we form the sum $V_N = \sum_{i=1}^{N} f_i \Delta R_i = \sum_{i=1}^{N} f_i A(R_i)$. If the number $N$ increases beyond all bounds and at the same time the greatest of the diameters of the subregions tends to zero, then $V_N$ tends to a limit $V$. This limit is independent of the particular nature of the subdivision of the region R...We call the limit $V$ the (double) integral.
However, they never provide any proof of the statement that "this limit is independent of the particular nature of the subdivision", and subsequent results are developed using the familiar assumption of "rectangular" partition.
Given how common it is in applied contexts (e.g. physics) to see calculations which do in fact assume that the limit is independent of the "shape" of the partition subsets (rectangular or not), I'm kind of surprised to not be able to find a proof of this (certainly it's intuitively very reasonable, but that's not the same thing as proof)
So my question is: does anyone know of a good resource (book, lecture notes, etc.) which does provide a proof that the Riemann integral of $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is independent of the (Jordan-measurable) "shape" of the subsets of the partition? (ideally with more relaxed continuity assumptions about $f$ than Courant & John's "continuous everywhere")
- Is there some "obvious" way to show it that I'm not seeing? (I suspect not, otherwise Courant & John surely would have mentioned it?)
- Maybe it follows most easily as a result of viewing Riemann integration as a special case of Lebesgue integration?
Footnote -- Sources with Riemann Integral via Strictly Rectangular Partitions:
- Apostol, Calculus
- Apostol, Mathematical Analysis
- Spivak, Calculus on Manifolds
- MIT OCW Lecture Notes