Exploring Meager Sets, Measure Zero, and Closed Sets with Empty Interior

In the realm of measure theory, the concepts of meager sets, sets of measure zero, and closed sets with empty interior are fundamental. Let's delve into these topics and understand the nuances of their relationships.

Meager Sets and Set of Measure Zero

In measure theory, a meager set (also known as a first category set) is a set that can be expressed as a countable union of nowhere dense sets. A nowhere dense set is one whose closure has an empty interior. Conversely, a set of measure zero is a set that can be covered by a countable union of intervals or more generally, sets whose total measure can be made arbitrarily small.

Key Insight: Not all meager sets have measure zero. While it is true that every set of measure zero is meager, the converse is not universally true. There exist meager sets that have positive measure. A classic example is the set of rational numbers in the interval [0,1]. This set is countable and thus meager, but it has a Lebesgue measure of zero.

Closed Sets with Empty Interior

A closed set having an empty interior is not necessarily of measure zero. This might seem counterintuitive, but there are counterexamples to illustrate this. A prominent example is the Cantor set. The Cantor set is a closed set with an empty interior, yet it has a measure of zero.

Another interesting example is the closed interval [0,1] minus the Cantor set. This set is closed and has a non-empty interior, and it has a positive measure. This demonstrates that the positive measure of a set is independent of its interior being empty or having a non-empty interior.

Constructing a Nowhere-Dense Set with Higher Measure

For 0 , there exists a nowhere-dense subset A of [0,1] such that lambda(A) geq 1 - varepsilon. This is a stronger statement and is a conceptually simple example of how to construct such a set. Here’s a step-by-step reasoning:

If A subset [0,1] does not contain any rationals, then its interior is empty.

If A is also a closed set, it is nowhere dense.

To obtain A by removing at most a set of measure varepsilon from [0,1], we fulfill the criteria lambda(A) geq 1 - varepsilon.

Constructing such a set involves the following observations:

There exists an open set U subset [0,1] containing mathbb{Q} cap [0,1] with lambda(U) leq varepsilon.

Then A : [0,1] setminus U is a nowhere-dense set and fulfills the criteria lambda(A) geq 1 - varepsilon.

Conceptual Understanding: The key here is that by carefully selecting a set U that covers the rationals but has a small measure, we can construct a set A that is both nowhere dense and has a large measure. This example helps to illustrate the distinction between the measures of sets and their topological properties.

Conclusion

While meager sets can often have measure zero, this relationship is not universally true. Similarly, a closed set with empty interior can have a positive measure, as demonstrated by the Cantor set and the example of [0,1] minus the Cantor set. Understanding these concepts is crucial for any deep dive into measure theory and set theory.