Rational Numbers Null Set under Lebesgue Measure

Theorem
Let $\lambda$ be $1$-dimensional Lebesgue measure on $\R$.

Let $\Q$ be the set of rational numbers.

Then $\lambda \left({\Q}\right) = 0$.

Proof
Let $\left({q_n}\right)_{n \in \N}$ be an enumeration of $\Q$.

It is ensured that such an enumeration exists by Rational Numbers are Countable.

As Lebesgue Measure is Diffuse, it holds that:


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

Also, the definition of $\left({q_n}\right)_{n \in \N}$ gives that:


 * $\displaystyle \bigcup_{n \mathop \in \N} \left\{{q_n}\right\} = \Q$

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


 * $\lambda \left({\Q}\right) = 0$