We denote by the Vitali set which is defined as follows:

*We say that are equivalent, and write , if and only if is a rational number. This equivalence relation partitions into an uncountable family of disjoint equivalence classes. By the axiom of choice there is a set which contains exactly one element from each equivalence class*.

Now let be a sequence of all rationals in with and define (mod 1).

Now we show that the are pairwise disjoint and

.

Indeed, if , then (mod 1) and (mod 1), with and belonging to . Consequently, , which means that and therefore . This shows that if . Since each is in some equivalence class, differs modulo 1 from an element in by a rational number, say , in . Thus , which proves that

.

The opposite inclusion is obvious.

**Question 1**. Show that there exist sets such that , and and

with strict inequality.

*Solution*. We put . Clearly, is a decreasing sequence. Since the are pairwise disjoint, we see that and . Moreover,

(the last inequality comes from the fact that is not measurable). It is now enough to show that

and the proof is complete.

**Question 2**. Show that there exist disjoint such that

with strict inequality.

*Solution*. We put then are pairwise disjoint and obviously

.

Moreover, all the are of the same outer measure. Thus which completes the proof.

**Question 3**. Show that each of the sets

is non-measurable.

**Question 4**. Show that if is a measurable subset of the Vitali set , then .

**Question 5**. Show that there exist sets and such that

but

**Question 6**. Show that any set of positive outer measure contains a non-measurable subset.