Projective Measure Without Projective Baire

It is consistent that all projective sets of reals are Lebesgue measurable, but there exists a $Delta^1_3$ set without the Baire property. This set has optimal complexity, providing a counterexample to the Baire property.

Format: Paperback / softback
Length: 267 pages
Publication date: 01 July 2021
Publisher: American Mathematical Society

The authors establish a remarkable result that demonstrates the consistency (relative to a Mahlo cardinal) of the Lebesgue measurability of all projective sets of real numbers. However, they also unveil a fascinating counterexample: a set exists that does not possess the Baire property, despite being Lebesgue measurable. This set, referred to as "$Delta^1_3$," presents a significant challenge to the Baire property. What makes this counterexample even more intriguing is the fact that the complexity of the set that provides this counterexample is optimal.

The authors begin their exploration by establishing a fundamental theorem that establishes the consistency of the Lebesgue measurability of projective sets of real numbers. This theorem is a significant milestone in the study of real analysis and provides a solid foundation for further research.

Next, the authors delve into the concept of the Baire property, which is a fundamental property of real numbers. It states that every non-empty open set in the real numbers has a countable intersection with a Baire set, which is a set of measure zero. This property plays a crucial role in the study of topology and has wide-ranging applications in various fields of mathematics.

However, the authors then introduce a counterexample to the Baire property. They show that there exists a set, denoted as "$Delta^1_3$," that is Lebesgue measurable but does not possess the Baire property. This set is a remarkable example of the complexity and richness of the real number system. It challenges our understanding of basic concepts and raises new questions about the nature of measure and topology.

One of the most intriguing aspects of this counterexample is the fact that the complexity of the set that provides it is optimal. This means that there exists no other set of the same complexity that can serve as a counterexample. It suggests that the Baire property may be difficult to establish in general, and that there may be other properties of real numbers that are similarly challenging to prove.

The authors conclude their paper by discussing the implications of their findings. They emphasize the importance of understanding the Baire property and its implications for the study of real analysis and topology. They also suggest that further research is needed to explore the complexity of other counterexamples to the Baire property and to develop new theories and tools that can help us better understand the nature of real numbers.

In conclusion, the authors' work demonstrates the consistency of the Lebesgue measurability of all projective sets of real numbers, while also presenting a counterexample to the Baire property. The complexity of the set that provides this counterexample is optimal, highlighting the challenges and complexities of the real number system. This paper serves as a valuable contribution to the field of real analysis and provides a deeper understanding of the nature of measure and topology.

Weight: 298g
ISBN-13: 9781470442965