Vitali Set Existence Theorem/Also presented as

Proof
The proof of the Vitali Set Existence Theorem is presented in using the following argument:

$M$ were Lebesgue measurable.

First, $\map \mu M = 0$ is impossible, because from $(1)$ this would mean $\map \mu \R = 0$.

Suppose $\map \mu M > 0$.

Then:

So $\map \mu M > 0$ is also impossible.

Hence $M$ is not Lebesgue measurable.

It needs to be pointed out that this:
 * $\ds \map \mu {\closedint 0 1} \ge \map \mu {\bigcup \set {M_r: r \in \Q \land 0 \le r \le 1} }$

is not immediately obvious, as it is not the case that:
 * $\ds \closedint 0 1 \supseteq \bigcup \set {M_r: r \in \Q \land 0 \le r \le 1}$

However, because $\map \mu X$ is translation invariant, $M_r$ has the same measure as $M_0$.

Hence the result.