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 ) 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.
|||P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802|
|||J.C. Oxtoby, "Measure and category" , Springer (1971) MR0393403 Zbl 0217.09201|
|||B.R. Gelbaum, J.M.H. Olmsted, "Counterexamples in analysis" , Holden-Day (1964) MR0169961 Zbl 0121.28902|
In certain models of ZF (not ZFC) every set of real numbers is Lebesgue measurable (a result of Solovay), so the axiom of choice is needed to construct a Lebesgue non-measurable set.
For Ulam's theorem see Cardinal number.
Non-measurable set. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Non-measurable_set&oldid=28255