Self-studying Rudin's RCA, and I want to make sure I am understanding the intricacies of his construction of the Lebesgue measure on $\mathbb{R}^n$.
The uniform continuity of $f$ shows that there is an integer $N$ and functions $g,h$ with support in $W$ such that: (i) $g,h$ are constant on each box in $\Omega_N$, (ii) $g \leq f \leq h$, (iii) $h-g < \epsilon$.
What is the simplest way to construct such functions? My first thought was to consider $\operatorname{cl}(A)$ of a box $A \in \Omega_N$ and apply the Extreme Value Theorem to obtain a maximal and minimal value of $f(x)$. Setting $h,g$ as those maximum and minimum values for $x \in A$, respectively, then gives (i),(ii). The Uniform Continuity Lemma then gives (iii) via an appropriate choice of $N$.
This however strikes me as a needlessly complicated construction; I am confident Rudin had something simpler in mind.
Property 2.19c then shows that: $$\forall n > N, \quad \Lambda_N g = \Lambda_n g \leq \Lambda_n f \leq \Lambda_n h = \Lambda_N h$$
This seems to follow if we partition $P_n$ into the boxes $P_n \cap A$. But once again it seems that some simpler methodology would be applicable.
Thus the upper and lower limits of $\{\Lambda_n f\}$ differ by at most $\epsilon \operatorname{vol}(W)$, and since $\epsilon$ was arbitrary we have proved the existence of
$$\lim\limits_{n \to \infty} \Lambda_n f = \Lambda f$$
Using the formalism described above, no natural way to establish this limit suggests itself, but this should be remedied by superior ways of establishing the above claims. More specifically, we obtain that
$$\lim\limits_{\epsilon \to 0} \;(\limsup\limits_{n \to \infty} \Lambda_n f - \liminf\limits_{n \to \infty} \Lambda_n f) = 0.$$ How precisely to we formally proceed from the above to equality of the inferior and superior limits, and therefore convergence? Any insight would be greatly appreciated.
For the first of the items you quoted, you don't need the Extreme Value Theorem. Use uniform continuity of $f$ to arrange for the boxes in $\Omega_N$ to be small enough so that $f$ does not vary by more than $\epsilon/10$ within any single box. Then, in each box, take the value of $f$ anywhere you like in the box, say at the center, and define $g$ (respectively $h$) throughout that box to be that value minus (respectively plus) $\epsilon/3$. (The $10$ was overkill, but I lack the energy, time, and desire to reduce it.)