Ngô Quốc Anh

August 15, 2009

Several questions involving the Vitali set

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.