I have this problem
For $\varepsilon>0$, let $G_\varepsilon$ be the class of all open oriented rectangles in $\mathbb{R}^n$ with diameter $<\varepsilon$ and let $\tau_\varepsilon(I)=$ volume of $I$ for $I\in G_\varepsilon$. Show that the measure $\tau_\varepsilon^*$ on $\mathbb{R}^n$ is the Lebesgue measure.
For any premeasure $\tau$, we define
$\tau^*(A)=\underset{\left\{C_i\right\}_{i=1}^\infty\subset G\\ A\,\subset\,\bigcup C_i}{\inf} \sum_i \tau(C_i)$
My problem is that i have no idea how to start this problem, I mean, isn't it obvious that any open interval in $\mathbb{R}$ can be split in smaller ones so that any open interval $A$ in $\mathbb{R}^n$ is complete covered by small oriented rectangles of diameter smaller than $\varepsilon$? Can anyone orient me on how to start this problem?
Thank you so much!