Lebesgue's measure $B=[0,1]\cap\mathbb{Q}$

Question

I have the following set: $$B=[0,1]\cap\mathbb{Q}$$ I want to get its Lebesgue's measure (that is, $\mu(B)$). My teacher says it's $0$ but I don't really get why. Could someone give me an explanation?

Answer 1

First of all, the measure of a set with a single element is clearly $0$. Indeed, for any $a\in\mathbb{R}$ we have $\{a\}\subseteq [a,a+\frac{1}{n})$ and by monotonicity $m(\{a\})\leq m([a,a+\frac{1}{n}))=\frac{1}{n}$. Since this is true for any $n\in\mathbb{N}$ it follows that $m(\{a\})=0$.

Now, your set $B$ is countable, hence you can write $B=(b_n)_{n=1}^\infty$. Then $B=\cup_{n=1}^\infty \{b_n\}$, this is a countable union of disjoint sets. By countable additivity:

$m(B)=\sum_{n=1}^\infty m(\{b_n\})=\sum_{n=1}^\infty 0=0$