Countable Set is Null Set under Lebesgue Measure

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Then $\lambda \left({S}\right) = 0$, 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:

$S = \displaystyle \bigcup_{n \mathop \in \N} \left\{{f \left({n}\right)}\right\}$


As Lebesgue Measure is Diffuse, it holds that:

$\forall n \in \N: \lambda \left({\left\{{f \left({n}\right)}\right\}}\right) = 0$

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

$\lambda \left({S}\right) = 0$

$\blacksquare$


Sources