Abstract:
In this paper, we construct a non-measurable subset of the Cantor set using an equivalence relation modulated by the rationals, similar to the construction of the classical Vitali set. We show that this subset retains many of the interesting properties of both the Cantor set and the Vitali set: it is uncountable, has measure zero, and is not Lebesgue measurable. This construction illustrates that even within sets of Lebesgue measure zero, non-measurable sets can be formed, highlighting the subtle interplay between set theory and measure theory.

1. Introduction

The concept of non-measurable sets has long intrigued mathematicians, with the Vitali set being one of the earliest known examples. Vitali's construction, reliant on the Axiom of Choice, demonstrates that the real numbers can be partitioned into equivalence classes under the relation of rational translation. By selecting one representative from each class, a non-measurable set is constructed.

On the other hand, the Cantor set is a well-known example of a compact, uncountable set that has Lebesgue measure zero. The Cantor set arises naturally in many contexts within analysis and topology, making it a canonical example of a "small" uncountable set.

In this paper, we combine these two ideas to construct a set that we will refer to as the Cantor-Vitali set. This set is a non-measurable subset of the Cantor set, combining the uncountability of both the Cantor and Vitali sets, while also inheriting the measure-zero property of the Cantor set.

2. Background

2.1 The Cantor Set

The Cantor set, ( C \subseteq [0,1] ), is constructed by successively removing the open middle third from intervals. Formally, begin with ( [0, 1] ), remove ( (1/3, 2/3) ), and iteratively continue removing the middle third of each remaining interval. After infinitely many steps, what remains is the Cantor set, which is:

  • Compact,
  • Perfect (every point is a limit point),
  • Uncountable,
  • Of Lebesgue measure zero.

2.2 The Vitali Set

The Vitali set, ( V \subseteq [0, 1] ), is constructed using the Axiom of Choice. Define an equivalence relation ( \sim ) on ( \mathbb{R} ) by:
[
x \sim y \iff x - y \in \mathbb{Q},
]
i.e., two real numbers are equivalent if their difference is rational. The Axiom of Choice is used to select one representative from each equivalence class. The resulting set ( V ) is:

  • Uncountable,
  • Not Lebesgue measurable,
  • Dense in ( [0, 1] ),
  • Of measure zero.

3. Construction of the Cantor-Vitali Set

We now proceed to construct the Cantor-Vitali Set, ( V_C ), by applying the equivalence relation from the Vitali set construction to the Cantor set.

3.1 Equivalence Relation on the Cantor Set

Consider the equivalence relation ( \sim ) defined on the Cantor set ( C \subseteq [0, 1] ) by:
[
x \sim y \iff x - y \in \mathbb{Q}.
]
This equivalence relation partitions ( C ) into equivalence classes, where each class contains points in ( C ) that differ by a rational number. Since ( C ) is uncountable and compact, these equivalence classes form a partition similar to that of the real numbers in the construction of the Vitali set.

3.2 Selection of Representatives

Using the Axiom of Choice, we select one representative from each equivalence class under ( \sim ). The set of these representatives forms the Cantor-Vitali Set, ( V_C ). Formally, we define:
[
V_C = { x \in C \mid x \text{ is the chosen representative from its equivalence class} }.
]
Since ( C ) is uncountable, so is ( V_C ). However, ( V_C \subseteq C ), and since ( C ) has measure zero, ( V_C ) also has measure zero.

4. Properties of the Cantor-Vitali Set

4.1 Non-Measurability

The set ( V_C ) is not Lebesgue measurable. The reasoning mirrors that of the classical Vitali set: if ( V_C ) were measurable, then the collection of rational shifts of ( V_C ) would cover ( C ), which has measure zero. However, summing up the measures of the shifted copies of ( V_C ) would lead to a contradiction, as these sets would have positive measure when viewed in aggregate. Hence, ( V_C ) must be non-measurable.

4.2 Measure Zero

Since ( V_C \subseteq C ) and ( C ) has Lebesgue measure zero, the set ( V_C ) also has measure zero:
[
\mu(V_C) = 0.
]
This is a crucial property, as it shows that non-measurable sets can exist even within sets of measure zero.

4.3 Uncountability

The set ( V_C ) is uncountable because it is derived from the uncountable Cantor set and contains one representative from each equivalence class under the rational translation relation.

5. Conclusion

The construction of the Cantor-Vitali set ( V_C ) illustrates the fascinating interplay between set theory and measure theory. By restricting the Vitali construction to the Cantor set, we obtain a non-measurable set with measure zero. This result highlights the complexity of measure theory and the subtle ways in which the Axiom of Choice can generate sets with unusual properties.

Further exploration could investigate the properties of such sets within other measure-theoretic contexts or explore similar constructions in higher-dimensional spaces.

References:

  • Vitali, G. Sul Problema della Misura dei Gruppi di Punti di una Retta, Bologna, 1905.
  • Halmos, P.R. Measure Theory, Springer, 1974.
  • Rudin, W. Principles of Mathematical Analysis, McGraw-Hill, 1976.