Countable Sets Have 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.