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:
- $\map \lambda \Q = 0$
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 $11 \ \text{(ii)}$