Rational Numbers form Null Set under Lebesgue Measure

Theorem

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

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

Then:

$\map \lambda \Q = 0$

that is, $\Q$ 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

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 4$: Problem $11 \ \text{(ii)}$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): almost all (almost everywhere)