Question. Is there a mathematical analysis textbook that contains a theorem justifying the procedure described below that is typical of calculus textbooks?
Many calculus books of several variables and even many physics and engineering textbooks use the procedure of decomposing a region $K$ of the plane $\mathbb{R}^2$ (respectively the space $\mathbb{R}^3$) into parts that are not necessarily rectangles $R_{k}$ in $\mathbb{R}^2$ (respectively parallelepiped $R_{k}$ in $\mathbb{R}^3$). For illustration purposes I have made the picture below for $\mathbb{R}^2$ setting.Pro
Formal description of the procedure. In general for $K\subset \mathbb{R}^{n}$ the procedure is nothing more than taking a collection $\mathscr{P}=\{A_{\ell}\subset K: 1\leq \ell\leq M\}$ with the following properties:
- for all $\ell$, the set $A_{\ell}$ is bounded, with non-empty and connected interior;
- for all $\ell$ the boundary $\partial A_\ell$ is of class $C^{1}$ piecewise;
- $\bigcup_{\ell}A_{\ell}=K$
- for all $\ell^{\,\prime}$ and $\ell^{\,\prime\prime}$ hold $ A_{\ell^{\,\prime}}\cap A_{\ell^{\,\prime\prime}} \subset \partial A_{\ell^{\,\prime}}\cap \partial A_{\ell^{\,\prime\prime}}\neq \emptyset$ or $ A_{\ell^{\,\prime}}\cap A_{\ell^{\,\prime\prime}}=\emptyset$.
Then, take for each $\ell$ a point $\zeta_{\ell}\in A_{\ell}$ and make the limit $$ \lim_{\|\mathscr{P}\|\to 0}\sum_{\ell=1}^{M}f(\zeta_\ell)\cdot {vol}(A_\ell), \qquad \mathrm{ with } \quad \|\mathscr{P}\|\overset{_{\mathrm{def}}}{=}\max_{1\leq \ell\leq M}\mathrm{diam}(A_{\ell}) $$ equal to the integral $$ \int_{K} f(x)\,\mathrm{d}x. $$
I am looking for elementary analysis textbooks that are not from Geometric Theory of Measurement or books that justify such a procedure with results from Lebesgue's Integral or Coarea's Formula.
I can construct and prove a theorem that justifies such a procedure in terms of the upper and lower sums of a Riemann integral. What I need is an analysis textbook in $\mathbb{R}^n$ space that proves a theorem that justifies such a procedure. My purpose is to cite the result in a paper I am writing.
On the other hand, mathematical analysis books such as Pugh, Rudin and Spivak do not present any theorems justifying such a procedure.