Countable Set is Null Set under Lebesgue Measure
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Theorem
Let $S \subseteq \R$ be a countable set.
Then $\map \lambda S = 0$, where $\lambda$ is Lebesgue measure.
That is, $S$ is a $\lambda$-null set.
Proof
By Surjection from Natural Numbers iff Countable, there exists a surjection $f: \N \to S$.
It follows that:
- $S = \ds \bigcup_{n \mathop \in \N} \set{\map f n}$
As Lebesgue Measure is Diffuse, it holds that:
- $\forall n \in \N: \map \lambda {\set{\map f n}} = 0$
Thus, by Null Sets Closed under Countable Union, it follows that:
- $\map \lambda S = 0$
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 10$: Problem $10.4$