Countable Set is Null Set under Lebesgue Measure

Theorem
Let $S \subseteq \R$ be a countable set.

Then $\map \lambda S = 0$, where $\lambda$ is Lebesgue measure.

That is, $S$ is a $\lambda$-null set.

Proof
By Surjection from Natural Numbers iff Countable, there exists a surjection $f: \N \to S$.

It follows that:


 * $S = \ds \bigcup_{n \mathop \in \N} \set{\map f n}$

As Lebesgue Measure is Diffuse, it holds that:


 * $\forall n \in \N: \map \lambda {\set{\map f n}} = 0$

Thus, by Null Sets Closed under Countable Union, it follows that:


 * $\map \lambda S = 0$