# Non-measurable set

A set that is not a measurable set. In more detail: A set belonging to a hereditary -ring is non-measurable if

here is the -ring on which the measure is given, and and are the exterior and interior measures, respectively (see Measure).

For an intuitive grasp of the concept of a non-measurable set, the following "effective constructions" are useful.

Example 1. let

be the unit square and define a measure on the sets

where runs through the Lebesgue-measurable sets of measure , by setting . Then the set

is non-measurable, since , .

The oldest and simplest construction of a non-measurable set is due to G. Vitali (1905).

Example 2. Let be the set of all rational numbers. Then a set (called a Vitali set) having in accordance with the axiom of choice exactly one element in common with every set of the form , where is any real number, is non-measurable. No Vitali set has the Baire property.

Example 3. Let (respectively, ) be the set of numbers of the form , where is an irrational number, and are integers with even (respectively, odd), and let be a set obtained by means of the axiom of choice from the equivalence classes of the set of real numbers under the relation:

Let . Then for every measurable set :

Yet another construction of a non-measurable set is based on the possibility of introducing a total order in a set having cardinality of the continuum.

Example 4. There exist a set such that and intersect every uncountable closed set. Any such set (a Bernstein set) is non-measurable (and does not have the Baire property). In particular, any set of positive exterior measure contains a non-measurable set.

Apart from invariance under a shift (Example 2) and topological properties (Example 3) there are also reasons of a set-theoretical character why it is impossible to define a non-trivial measure for all subsets of a given set; such is, for example, Ulam's theorem (see [2]) for sets of bounded cardinality.

No specific example is known of a Lebesgue non-measurable set that can be constructed without the use of the axiom of choice.

#### References

 [1] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802 [2] J.C. Oxtoby, "Measure and category" , Springer (1971) MR0393403 Zbl 0217.09201 [3] B.R. Gelbaum, J.M.H. Olmsted, "Counterexamples in analysis" , Holden-Day (1964) MR0169961 Zbl 0121.28902