Finding the outer measure of the x-axis in $\mathbb{R}^2$
Solution 1:
Take the following set of rectangles:
$$Q_k=[k,k+1]\times\left[-\frac{\epsilon}{2^{|k|}},\frac{\epsilon}{2^{|k|}}\right] \;\;\text{ for each }\;\; k\in \mathbb{Z}$$
$$m_*(E)\leq \sum\limits_{k\in \mathbb{Z}}|Q_k|= 2\epsilon + 2\sum_{k\geq 1}\frac{\epsilon}{2^{k-1}}=6\epsilon$$
And you can take $\epsilon>0$ arbitrarily small, that is to say, that $m_*(E)\leq 0$.
As you can see, the trick is to make each covering's measure finite and dependant of a free parameter, in this case $\epsilon$.