Why do the integers, rationals and any countable set have zero measure?

Question

There is an exercise in my text that tells me to prove the "obvious and easy to see" fact that $\mathbb{Z}$ and $\mathbb{Q}$ have measure zero.

Er...here is what I know so far. If I have an interval, then the measure is the end point subtracting the initial point i.e. the length of that interval. What can I do to extend this line of thinking to $\mathbb{Z}$ and $\mathbb{Q}$ and any countable set?

Answer 1

This comes from the fact that a measure is countably additive.

Let $I$ be a countable set. You have then

$$\mu(I) = \mu ( \bigcup_{x\in I} \{x\} ) = \sum_{x\in I} \mu( \{x\} ) $$

Now if every singleton has measure zero, it follow that

$$\mu(I) = 0$$

And this is the case for the Lebesgue's measure on the real line.

Answer 2

Hints: enumerate any countable subset of $\mathbb R$ as, say, $\{a_n\}_n \ge 1$. Now cover $a_n$ by an interval of length $\frac{\varepsilon}{2^n}$. What is the total measure of the cover? Notice that $\varepsilon>0$ may be chosen arbitrarily. What does that imply?