How to get the inner measure of the Vitali set.

Question

I know that this should be very easy, but why exactly is the inner measure of the Vitali set 0?

Answer 1

Note that for any collection of disjoint sets $\{U_n\}_{n \in \mathbb{Z}}$, we have $$ \mu_*(\bigcup_{n \in \mathbb{N}} U_n) \geq \sum_{n \in \mathbb{N}}\mu_*(U_n) $$ Now, if we take $U_n$ to be a covering of $[0,1]$ under the usual construction, we find that $\mu_*(U_n)$ is the same for each $n$. Let $M$ be this value, then $$ \mu_*(\bigcup_{n \in \mathbb{N}} U_n) \geq \sum_{n \in \mathbb{N}}\mu_*(U_n) = \sum_{n \in \mathbb{N}} M $$ This can only be true if $M = 0$.