Countable Set has Measure Zero

Theorem
If $$X \ $$ is a countable set, then the measure of $$X \ $$ is $$m \left({X}\right) = 0 \ $$.

Proof
Let $$\displaystyle \left\{{x_i}\right\}_{i=1}^\infty \ $$ be an enumeration of the elements of $$X \ $$.

For any positive number $$\epsilon \ $$, define

$$A_i = \left({x_i - 2^{-i}\epsilon, x_i + 2^{-i}\epsilon}\right) \ $$.

Then $$\displaystyle X \subseteq \bigcup_{i=1}^\infty A_i \ $$ and $$\displaystyle m \left({\bigcup A_i}\right) \le \sum_{i=1}^\infty 2^{1-i}\epsilon = 2\epsilon \ $$.

Since our choice of $$\epsilon \ $$ was arbitrary, for any positive real $$z \ $$ we can construct a set $$Y \ $$ such that $$X \subseteq Y \ $$ and $$m \left({Y}\right) \leq z \ $$. Hence $$X \ $$ has zero measure.