Non-measurable set

A set that is not a measurable set. In more detail: A set $X$ belonging to a hereditary $\sigma$- ring $H ( S)$ is non-measurable if

$$\mu ^ {*} ( X) > \ \mu _ {*} ( X) ;$$

here $S$ is the $\sigma$- ring on which the measure $\mu$ is given, and $\mu ^ {*}$ and $\mu _ {*}$ 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

$$K = \{ {( x , y ) } : {0 \leq x \leq 1 , 0 \leq y \leq 1 } \}$$

be the unit square and define a measure $\mu$ on the sets

$$\widehat{E} = \{ {( x , y ) } : {x \in E , 0 \leq y \leq 1 } \} ,$$

where $E$ runs through the Lebesgue-measurable sets of measure $m ( E)$, by setting $\mu ( \widehat{E} ) = m ( E)$. Then the set

$$X = \{ {( x , y ) } : {0 \leq x \leq 1 , y = 1 / 2 } \}$$

is non-measurable, since $\mu ^ {*} ( X) = 1$, $\mu _ {*} ( X) = 0$.

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

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

Example 3. Let $B$( respectively, $C$) be the set of numbers of the form $n + m \xi$, where $\xi$ is an irrational number, $m$ and $n$ are integers with $n$ even (respectively, $n$ odd), and let $X _ {0}$ be a set obtained by means of the axiom of choice from the equivalence classes of the set of real numbers under the relation:

$$x \sim y \ \textrm{ if } \ x - y \in A = B \cup C .$$

Let $X = X _ {0} + B$. Then for every measurable set $E$:

$$\mu _ {*} ( X \cap E ) = 0 ,\ \ \mu ^ {*} ( X \cap E ) = \mu ( E) .$$

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 $B \subset \mathbf R$ such that $B$ and $\mathbf R \setminus B$ 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