Vitali set

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In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, named after Giuseppe Vitali. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence is proven on the assumption of the axiom of choice.

Contents 1 Measurable sets 2 Construction and proof 3 See also 4 References

[edit] Measurable sets

Certain sets have a definite 'length' or 'mass'. For instance, the interval [0, 1] is deemed to have length 1; more generally, an interval [a, b], a ≤ b, is deemed to have length b − a. If we think of such intervals as metal rods with uniform density, they likewise have well-defined masses. The set [0, 1] ∪ [2, 3] is composed of two intervals of length one, so we take its total length to be 2. In terms of mass, we have two rods of mass 1, so the total mass is 2.

There is a natural question here: if E is an arbitrary subset of the real line, does it have a 'mass' or 'total length'? As an example, we might ask what is the mass of the set of rational numbers, given that the mass of the interval [0, 1] is 1. The rationals are dense in the reals, so any positive value between 0 and 1 may appear reasonable.

However, it turns out that the closest analogue to mass is the Lebesgue measure, which assigns a measure of b − a to the interval [a, b], but will assign a measure of 0 to the set of rational numbers. Any set which has a well-defined Lebesgue measure is said to be "measurable", but the construction of the Lebesgue measure (for instance, using the outer measure) does not make it obvious whether there exist non-measurable sets.

[edit] Construction and proof

If x and y are real numbers and x − y is a rational number, then we write x ~ y and we say that x and y are equivalent; ~ is an equivalence relation. For each x, there is a subset [x] = {y in R : x ~ y} of R called the equivalence class of x. The set of these equivalence classes partitions R. By the axiom of choice, we are able to choose a set containing exactly one representative out of each equivalence class (for any equivalence class [x], the set V ∩ [x] is a singleton). We say that V is a Vitali set.

A Vitali set is non-measurable. To show this, we assume that V is measurable. From this assumption we use two properties of Lebesgue measurable sets and prove something absurd: namely that a + a + a + ... (an infinite sum of identical numbers) is between 1 and 3. Since an absurd conclusion is reached, it must be that the only unproved hypothesis (V is measurable) is at fault.

First we let q₁, q₂, ... be an enumeration of the rational numbers in [−1, 1] (recall that the rational numbers are countable). From the construction of V, note that the sets , k = 1, 2, ... are pairwise disjoint, and further note that . (To see the first inclusion, consider any real number x in [0,1] and let v be the representative in V for the equivalence class [x]; then x −v = q for some rational number in [-1,1] (say q = q_l) and so x is in V_l.)

From the definition of Lebesgue measurable sets, it can be shown that all such sets have the following two properties:

1. The measure is countably additive, that is if A_i is a set of at most a countable number of pairwise-disjoint sets, then

2. The measure is translation invariant, that is, for any real number x, μ(A) = μ(A + x).

Consider the measure μ of the inclusion equation above and apply the above two properties.

Because μ is countably additive, it must also have the property of being monotone; that is, if A⊂B, then μ(A)≤μ(B). Hence, we know that

By countable additivity, one has

with equality following because the V_k are disjoint. Because of translation invariance, we see that for each k = 1, 2, ..., μ(V_k) = μ(V). Combining this with the above, one obtains

The sum is an infinite sum of a single real-valued constant, non-negative term. If the term is zero, the sum is likewise zero, and hence it is certainly not greater than or equal to one. If the term is nonzero then the sum is infinite, and in particular it isn't smaller than or equal to 3.

So the above conclusion is absurd, and since all we've used is translation invariance and countable additivity, it must be true that V is non-measurable.

[edit] See also

• Non-measurable set
• Banach–Tarski paradox

[edit] References

• Herrlich, Horst: Axiom of Choice, page 120. Springer, 2006.
• Vitali sets at Finance wiki Retrieved 03-10-2009