Demonstrating the Measurability of a Set with Zero Lebesgue Outer Measure

Understanding and demonstrating the measurability of a set in measure theory, particularly when the set has a Lebesgue outer measure of zero, is a fundamental concept. In this article, we will explore the various approaches to proving that such a set is indeed measurable. We will analyze the relationship between inner and outer measures, and how they combine to establish the measurability of the set.

Introduction to Lebesgue Measure and Outer Measure

Lebesgue measure is a fundamental concept in real analysis that generalizes the notion of length, area, and volume. Specifically, the Lebesgue outer measure is a number assigned to every subset of the real numbers, providing an upper bound for the size of the set.

Measurability Criteria

A set E in the real numbers (?^n) is considered Lebesgue measurable if for every subset A of ?^n, the following holds:

μ*(A) μ*(A ∩ E) μ*(A ∩ Ec)

where μ* denotes the Lebesgue outer measure of a set. An equivalent definition of measurability can be formulated using the concept of inner and outer measures.

Inner and Outer Measure Relationship

For a given set E, the inner measure, denoted μ*(E), is defined as the supremum of the measures of all compact subsets of E. The outer measure, denoted μ*(E), is defined as the infimum of the measures of all open sets that contain E. When the inner and outer measures of a set are equal, the set is considered measurable.

Key Theorem: When Outer Measure is Zero, the Set is Measurable

One of the fundamental theorems in measure theory is that a set with zero Lebesgue outer measure is measurable. Here is a detailed proof of this theorem:

Assumption: Let E be a subset of ? with Lebesgue outer measure zero, i.e., μ*(E) 0.

Using the definition of outer measure, for every ε > 0, there exists a countable collection of open intervals {(In)} such that E ? ∪n In and ∑n |In| ε, where |In| represents the length of interval In.

Next, we use the property that for any set E with Lebesgue outer measure zero, the inner measure of E must also be zero. This is because for any subset A of ?, the outer measure of E is less than or equal to the outer measure of A ∩ E, which implies that μ*(A ∩ E) 0 for all A.

By the definition of inner measure, μ*(E) 0 means that for every δ > 0, there exists a compact set K such that E ? K and μ(K) .

Since the inner measure and outer measure are equal for E, we conclude that μ*(E) μ*(E) 0, which confirms that E is measurable.

Alternative Proofs and Developments

There are alternative ways to prove that a set with zero Lebesgue outer measure is measurable, depending on the developmental approach to measure theory. Here are two such approaches:

Approach via Inner Measure and Outer Measure

In the approach that uses both inner and outer measures, one can directly show that for any measurable set E in ?, 0 ≤ μ*(E) ≤ μ*(E). Given this inequality, if μ*(E) 0, it follows immediately that μ*(E) 0 and hence E is measurable.

Approach via Outer Measure Alone

A different approach focuses solely on outer measures. In this case, one can show that for any set E with Lebesgue outer measure zero, there exists a set H of type g-δ (a countable intersection of open sets) such that E ? H and μ(H) 0. Since μ(H) 0, it follows that μ*(E) 0, and thus E is measurable.

Conclusion

Proving that a set with zero Lebesgue outer measure is measurable is a cornerstone of measure theory and is crucial for understanding more advanced concepts. Whether through the comparison of inner and outer measures or the use of outer measures alone, the result holds true, providing a rigorous foundation for further exploration in real analysis and measure theory.