countable sets have outer measure zero

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Note

The example from outer measure also means that any countable $A \subset R$ has $m^{*} (A) = 0$ .

Example

$m^{*} (Q) = 0$

Proof

If $A$ is countably infinite, then we can write $A = {a_{1}, a_{2}, \dots} = {a_{n}}_{n \in N}$

Let $ϵ > 0$ . We will show $m^{*} (A) \leq ϵ$ . For each $n \in N$ , let $I_{n} = (a_{n} - \frac{ϵ}{2^{n + 1}}, a_{n} + \frac{ϵ}{2^{n + 1}})$

Then

\begin{aligned} A & \subset ⋃_{n} I_{n} \\ ⟹ m^{*} (A) & \leq \sum_{n = 1}^{\infty} ℓ (I_{n}) \\ = \sum_{i = 1}^{\infty} \frac{ϵ}{2^{n}} \\ = ϵ \end{aligned}

And since $ϵ$ was arbitrary, as $ϵ \to 0$ we get $m^{*} (A) = 0$ .

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References

countable sets have outer measure zero

References

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