# 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 $\lambda \left({\Q}\right) = 0$, i.e. $\Q$ is a $\lambda$-null set.

## Proof

We have that the Rational Numbers are Countably Infinite.

The result follows from Countable Set is Null Set under Lebesgue Measure.

$\blacksquare$