Measure theory Vitali nonmeasurable set.

Question

On $[0,1]$ I've got the relation $\sim$ defined as:

$x \sim y \iff x-y \in \mathbb Q$

this is a relation of equivalence and so: we can make a factor class

$$[0,1]|_{\sim}=\{[x]|x\in[0,1],y\in[x]\iff x \sim y\}$$

then we make, using the axiom of choice:

$$A=\{x|x-\text{ is the representative of class $[x]$}\}$$

Then we take $\mathbb Q \cap [-1,1]=\{r_1,r_2,...\}$ and lets evaluate the family $\{(r_i+A)\}$

My question is the resoning behind $$[0,1]\subseteq \bigcup_{j=1}^{\infty}(r_j+A)$$

The proof it gives is that: Let $x \in [0,1],$ then there exists $y\in A$ such that $x\in [y]$ or $x-y=r_j \in \mathbb Q...$ (why does there a $y$ like this, formally looking at it???)

Answer 1

Evidently $x\in[x]$.

Now let $y$ be the representative of $[x]$ that belongs to $A$.

Then $x\in[x]=[y]$.

So $x-y\in\mathbb Q$ and since $x,y\in[0,1]$ we find: $$x-y\in[-1,1]\cap \mathbb Q=\{r_1,r_2,\dots\}$$