Content $0$ implies boundary has content $0$

Problem: If a set $C$ has content $0$, then the boundary must also have content $0$.

The solution say

Solution: Suppose a finite set of rectangle $ U_i = (a_{i1},b_{i1}) \times \dots \times (a_{in}, b_{in})$ where $i \in \{1, \dots, m \}$ cover $C$ and have total volume less than $\epsilon/ 2$ where $\epsilon >0.$ Let $V_i = (a_{i1} - \delta ,b_{i1} + \delta) \times \dots \times (a_{in} - \delta , b_{in} + \delta)$ where

$$\prod_{i\in \{ 1,\dots,n\}} (b_{ij} - a_{ij} + 2\delta) - \prod_{i\in \{ 1,\dots,n\}} (b_{ij} - a_{aj} ) < \epsilon/2m .$$

Or equivalently,

Then the union of $V_i$ cover the boundary of $C$ and have total volume less than $\epsilon$. Hence $\partial C$ is also of content $0$.

This is a really simple question, but does $V_i - U_i$ cover $\partial C$? $V_i - U_i$ means the part where we remove $U_i$ from $V_i$.

Also is the last step because of $$ \sum_{i = 1}^m vol(\cup( V_i - U_i)) \leq \sum_{i = 1}^m\prod (b_{ij} - a_{aj} + 2\delta) - \prod (b_{ij} - a_{aj} ) < \sum_{i = 1}^m \epsilon/2m = \epsilon /2.$$

or equivalently,

$$ \sum_{i = 1}^m vol(\cup V_i ) \leq \sum_{i = 1}^m\prod (b_{ij} - a_{aj} + 2\delta) < \sum_{i = 1}^m \epsilon/2m + \prod (b_{ij} - a_{aj} ) < \epsilon /2 + \epsilon/2 = \epsilon. $$

Am I also correct to understand the outline of this proof is

1. Use the finite cover that covers $C$, and take a $\delta$ cover over that finite cover just enough to cover the boundary of $C$.

2. Using the finite cover of $C$, demand this cover so that it has volume less than $\epsilon/2$. Pick $\delta$ small enough so that we get

$$\prod_{i\in \{ 1,\dots,n\}} (b_{ij} - a_{ij} + 2\delta) - \prod_{i\in \{ 1,\dots,n\}} (b_{ij} - a_{aj} ) < \epsilon/2m .$$

1. Use finite subadditivitly to finish off the estimation.

$V_i-U_i$ will not cover $C$ in general. Note that $U_i$ is large enough to cover $C$, but it may be wasteful. So the only trick is really to start with a sufficiently small finite cover for $C$, and blow it up slightly so that $V_i\supset \overline {U_i}$ and hence $\bigcup V_i\supset \bigcup \overline{U_i}=\overline{\bigcup U_i}\supset \overline C\supset \partial C$, and thus obtain an arbitrarily small cover of $\partial C$.