Let $A=[x_1-r,x_1+r]\times \cdots \times[x_n-r,x_n+r]\subset \mathbb{R}^n$, for $x_i\in \mathbb{R}$ and $r>0$.
If $S\subset D\subset \mathbb{R}^n$ has content zero, such that $D$ is open and non-empty, prove for each $\varepsilon> 0$, there are $A_1,\ldots,A_n\subset D$, such that
$$S\subset A_1\cup \cdots \cup A_n \text{ and } \sum_{j=1}^n\mu(A_j)<\varepsilon$$
I know the Jordan content/measure of $A$ is $\mu(A)=\prod_{i=1}^N(x_1+r-(x_1-r))=2^Nr^N$, and that $Z$ is open, which implies for each $\varepsilon>0$, there are compact intervals $I_1,\ldots,I_n\subset \mathbb{R}^n$, such that $Z\subset \bigcup_{I=1}^nI_i$ and $\sum_{I=1}^n\mu(I_i)<\varepsilon$. Furthermore, I need to show these compact intervals $I_i$ can be modelled as $A_i$ as defined above. This is where I am stuck. I was thinking of cutting each compact interval of $Z$ into cubes of the form $A$, and show it is less than some multiple of $\varepsilon$, but am not sure how to formally show that.
In addition, I define a set $S$ to have content zero, if for each $\varepsilon > 0$, there are compact intervals $I_1,\ldots,I_n\subset \mathbb{R}^n$, such that $S\subset \bigcup_{i=1}^n I_i$, and $\sum_{I=1}^n \mu(I_i)< \varepsilon$
I am not sure how to attempt the above problem, and would appreciate any help if possible. Thanks in advance!
Since $S$ has content zero it is covered by compact intervals $I_1, I_2, \ldots, I_m$ with $\sum_{k=1}^m \mu(I_k) < \epsilon.$
There are three steps to the proof that there are a finitely many intervals $A_1, A_2, \ldots, A_n \subset D$ such that $S \subset \bigcup_{j=1}^n A_j$ and $\sum_{j=1}^m \mu(A_j) < \epsilon$.
(1) Show that for any open set $D \subset \mathbb{R}^N$ there is a sequence of increasing compact subsets $C_1,C_2, \ldots \subset D$ such that $C_j \subset \text{int} C_{j+1}$ and $D = \cup_{j \geqslant 1} C_j.$
(2) Using (1) show that there exists a countable collection of intervals $R_1, R_2, \ldots$ contained in $D$ such that $D \subset \bigcup_{j\geqslant 1} \text{int}R_j$.
(3) The intersection of two intervals can be decomposed into a finite number of intervals. Thus forming the intersection of the (finitely many) intervals $I_k$ with intervals $R_j$ we obtain the desired intervals $A_1, A_2, \ldots A_n$.
For (1) take $C_j = \bar{B}(0,j) \, \cap \, \{X: d(x,D^c) \geqslant j^{-1} \}$, the intersection of the closed ball of radius $j$ and the set of all point that are more than a distance of $j^{-1}$ from the complement of $D$.
Proof of (2) is given in Lemma 16.2 of Analysis on Manifolds by Munkres.