Examples of Non-Measurable Sets in Lebesgue Measure Theory

1. Vitali Set

One of the most famous examples of a non-measurable set is the Vitali set. The construction begins by considering the unit interval [0, 1].
We define an equivalence relation on ℝ such that two real numbers x and y are equivalent if x - y
is a rational number (i.e., x - y ∈ ℚ). Using the axiom of choice, we can select exactly one representative from each equivalence class.

The resulting set is called a Vitali set, and it can be shown that this set is non-measurable. This is because the translation invariance of Lebesgue measure
would lead to a contradiction if such a set were measurable.

2. Axiom of Choice-based Subsets of the Real Line

More generally, any set whose existence relies on the axiom of choice and does not have a constructive definition (such as the Vitali set) is likely to be non-measurable.
These sets are often subsets of ℝ or other uncountably infinite sets where elements are chosen based on arbitrary selection rather than a definable rule.

3. Bernstein Set

A Bernstein set is a subset of ℝ such that both the set and its complement intersect every uncountable closed subset of ℝ,
but neither contains any uncountable closed subset entirely. The Bernstein set can be shown to be non-measurable.

Its non-measurability stems from the fact that it cannot satisfy the conditions for Lebesgue measure. Intuitively, there’s no way to assign a consistent measure to such a set
given the intricate way it interacts with closed sets.

4. Hamel Basis of ℝ over ℚ

A Hamel basis is a basis for the real numbers ℝ considered as a vector space over the rationals ℚ.
The elements of a Hamel basis are such that every real number can be expressed uniquely as a finite linear combination of these elements with rational coefficients.

A Hamel basis is non-measurable because it can be used to translate a measurable set in ways that violate the translation invariance of the Lebesgue measure.
Essentially, shifting the basis elements by rational numbers can lead to contradictions in measure theory.

5. Example Based on the Banach-Tarski Paradox

While the Banach-Tarski paradox primarily involves 3-dimensional spaces, its construction hints at the existence of non-measurable sets.
The paradox shows that a solid ball in 3D space can be decomposed into a finite number of non-measurable pieces, which can then be reassembled into two balls identical to the original.

The sets used in this decomposition are non-measurable under Lebesgue measure.

In summary, non-measurable sets often arise from constructions that require the axiom of choice and exhibit strange properties, particularly with respect to translation and countable additivity. The Vitali set and Hamel basis are classic examples of such sets.

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