Lebesgue Measure and Non-Measurable Sets
By Schinella D’Souza
Published September 10th, 2019
In 1902, Lebesgue developed his theory of measure and integration. He was seeking a way to generalize the concepts of length, area, and volume, to n dimensions. His generalization is called Lebesgue measure. We will look at how to define measure and measurable sets and how to construct a set that is not measurable. We will also see some consequences of non-measurable sets.
2 Lebesgue Measure
To define Lebesgue measure, we follow the approach of Stein and Shakarchi. Another equivalent definition can be found in Royden and Fitzpatrick.
Definition 2.1 The outer measure of
is defined by
where the infimum is taken over all countable coverings
by closed cubes.
Definition 2.2 A set
is Lebesgue measurable if for every ε > 0, there exists an open set O ⊇ A such that m∗(O \ A) ≤ ε. The Lebesgue measure of A is m(A) = m∗(A).
The following three properties are consequences of Definition 2.2:
Countably Additive: If
are disjoint measurable subsets of
then we have that
2. Translation Invariant: Translating a set
preserves its Lebesgue measure.
3. Rotation Invariant: Rotating a set
preserves its Lebesgue measure.
3 Construction of a Non-Measurable Set
The natural question that follows from the definition of Lebesgue measure is if all sets are mea- surable. In 1905, Vitali showed that it is possible to construct a non-measurable set. The steps in the construction are as follows:
Define an equivalence relation ∼ on the set[0,1]: Let x ∼ y if x − y ∈ Q.
Denote the equivalence classes by
for α in some index set J. Then [0,1] =
Use the axiom of choice to choose precisely one element from each equivalence class and let the set of all these elements be called V . V is called a Vitali set.
Theorem 3.1 The Vitali set V is not measurable. Proof Outline of 3.1:
Suppose (for contradiction) that V is measurable and enumerate the rational numbers in the set [−1,1]. Denote this enumeration by
2. Consider the translations of the set V, that is,
These translations are disjoint and measurable. Moreover,
for all j ∈ N by translation invariance of V .
3. It can then be shown that
From the translation invariance and countable additivity properties, we then have the inequality:
4. If m(V ) = 0, then the inequality
would be violated. At the same time, this inequality implies that the measure of V is not positive. As the measure of V cannot be negative, we have a contradiction. Therefore, V is not measurable.
4 Dependence on the Axiom of Choice
The existence of sets that are not measurable relies on the acceptance of the axiom of choice. Accepting the axiom of choice leads to paradoxes, one of the most famous being the Banach-Tarski paradox. This paradox says that the three-dimensional unit ball can be decomposed into a finite number of disjoint sets and then reassembled into two unit balls. A similar formulation of this paradox is the pea and the sun paradox. This paradox says that a pea can be decomposed into pieces and rearranged to be as big as the sun. See  for more details on this.
Rejecting the axiom of choice has its consequences as well. For instance, Solovay  showed in 1970 that in a certain model of Zermelo-Fraenkel set theory excluding uncountable choice, every subset of real numbers is measurable. An implication of this would then be that every function can be integrated.
5 After Thoughts
Notice that the outer measure of a set is always defined. What is the outer measure of the Vitali set V we constructed? It cannot be 0 or 1, but has to be between 0 and 1. There is not enough information to determine the outer measure in our particular construction. Now think about if there is another way to construct a non-measurable set with any given outer measure. This is possible but will be left to the reader to think about. The references below delve deeper into the points we have touched upon.
 H.L. Royden and P.M. Fitzpatrick, Real Analysis, 4th ed. Location, state: Prentice Hall, 2010.
 E.M. Stein, R. Shakarchi, Real Analysis: Measure Theory, Integration, & Hilbert Spaces. Princeton, NJ: Princeton University Press, 2005.
 R.M. Solovay, “A Model of Set-Theory in Which Every Set of Reals is Lebesgue Measurable,”Annals of Mathematics, vol. 92, no. 1, p. 1 - 56, July 1970.
 B. Sury, “Unearthing the Banach-Tarski Paradox,” Indian Statistical Institute, October 2017. [Online], Available: https://www.ias.ac.in/article/fulltext/reso/022/10/0943-0953. [Accessed Sept. 18, 2007].
 L.M. Wapner, The Pea and the Sun: A Mathematical Paradox. Wellesley, MA: A K Peters, Ltd., 2005.