Rational Numbers Null Set under Lebesgue Measure

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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.

$\blacksquare$


Sources