Proving the Lebesgue Measure of Irrational Numbers in the Interval [0, 1]

When dealing with advanced mathematical concepts such as Lebesgue measure, it is essential to understand the definitions and properties clearly. In this article, we will explore the reasoning behind the Lebesgue measure of the set of irrational numbers within the interval [0, 1]. We will clarify the common misunderstanding and then proceed to provide a direct proof from the definition.

The Common Misunderstanding and Correction

Sometimes, there can be confusion regarding the existence of a set in the interval [0, 1] that is not Lebesgue measurable. However, it is important to note that there are no irrational numbers in the set {0.1}, as it is an empty set, and hence has Lebesgue measure zero. The correct statement to consider is whether the set of irrational numbers in the interval [0, 1] has Lebesgue measure zero.

Understanding the Definitions and Proofs

First, let's define the key concepts involved:

Locus of Irrational Numbers in [0, 1]

Let ( A [0,1] cap mathbb{Q} ) and ( B [0,1] setminus mathbb{Q} ).

By definition, the Lebesgue measure of the interval [0, 1] is given by:

[ mu [0,1] inf{b - a mid [0,1] subseteq [a, b]} 1 ]

Lebesgue Measure of Set A

Next, let's prove that ( mu A 0 ). Since ( A ) is the set of rational numbers in the interval [0, 1], and the rational numbers are countable, we can use the property of countable additivity of Lebesgue measure. Specifically, every countable subset of the real line has Lebesgue measure zero. Therefore, ( mu A 0 ).

Implication for Set B

Given that ( mu A 0 ), by the countable additivity property of the Lebesgue measure, we have:

[ mu B mu [0,1] - mu A 1 - 0 1 ]

Hence, the set of irrational numbers in [0, 1] has Lebesgue measure 1.

Exploring the Theorems and Proofs Further

There are two fundamental theorems in measure theory that are relevant here:

Theorem 1: Every Subset of the Set of Rationals is Measurable of Measure Zero

This theorem states that any subset of (mathbb{Q}) is Lebesgue measurable and has measure zero. Since ( A [0,1] cap mathbb{Q} ) is a subset of (mathbb{Q}), it follows that ( mu A 0 ).

Theorem 2: The Disjoint Union of a Measurable and a Non-Measurable Set is Non-Measurable

If we assume the existence of a non-measurable subset of [0, 1], then the union of this set with a measurable set (the irrationals in this case) would also be non-measurable. However, we have already shown that the set of irrationals has measure 1, hence it cannot be non-measurable. This contradiction implies that the set of irrationals in [0, 1] must be measurable.

Conclusion

In summary, the set of irrational numbers in the interval [0, 1] has Lebesgue measure 1. This result is a consequence of the countable additivity property of the Lebesgue measure and the properties of the rational and irrational numbers within the interval. It is crucial to understand these foundational concepts in measure theory to accurately analyze sets and their measures.