Exploring the Possibility of a New Measure Theory Where All Sets Are Measurable

The concept of measurable sets is fundamental in modern mathematics, particularly in analysis, probability, and measure theory. However, the existence of nonmeasurable sets, a classic result derived under the assumption of the Axiom of Choice, has sparked considerable debate and interest in the possibility of creating an alternative measure theory where every set is measurable. This article delves into the current understanding of this concept, focusing on the Solovay model as a significant breakthrough in this area.

The Axiom of Choice and Nonmeasurable Sets

The Axiom of Choice is one of the most controversial and well-known axioms in set theory. It states that for any collection of nonempty sets, it is possible to choose one element from each set. While widely accepted in most mathematical contexts, it has far-reaching implications that sometimes challenge our intuition. One of these implications is the existence of nonmeasurable sets. In the standard model of set theory, Zermelo-Fraenkel set theory (ZF) combined with the Axiom of Choice (ZFC), there exist subsets of the real numbers that are not Lebesgue measurable.

The nonmeasurability of certain sets, such as the Vitali set, is deeply connected to the Axiom of Choice. These sets defy the intuitive notion of a "size" or "volume" in the real numbers and have profound consequences for measure theory. However, the question arises: Can we create a new measure theory where all sets are measurable? The answer, surprisingly, is affirmative, at least in certain models of set theory.

The Solovay Model

Robert M. Solovay’s model of set theory is a remarkable achievement in this context. In Solovay's model, denoted as ZF DC (the Axiom of Dependent Choice), it is possible to construct a "measurable" version of the real numbers where every subset of the real numbers is Lebesgue measurable. This model is significant because it shows that the existence of nonmeasurable sets is not a universal consequence of the ZFC axioms but rather a specific result derived from the Axiom of Choice.

In the Solovay model, the Axiom of Choice is replaced by the weaker Axiom of Dependent Choice, and the Continuum Hypothesis (CH) is negated. This model ensures that all sets of reals are Lebesgue measurable, and every set of reals has the property of Baire and the perfect set property. These properties are crucial in measure theory and provide a coherent framework where the intuitive notion of a "size" for subsets of the real numbers is consistent and well-defined.

Implications and Challenges

The Solovay model, while mathematically elegant, does not come without its challenges. One of the primary concerns is the loss of some of the structural results that are typically derived from the full Axiom of Choice. For example, in the Solovay model, there does not exist a well-ordering of the real numbers, and the Banach-Tarski paradox, which relies on the Axiom of Choice, does not hold.

Another challenge is the computational and practical aspects of working within this model. While the theoretical guarantees are strong, the transition to a model where all sets are measurable can complicate various constructions and calculations that are more straightforward in the ZFC model. Nonetheless, the Solovay model and related developments provide a valuable framework for exploring alternative measure theories and the foundations of mathematical analysis.

Conclusion

The possibility of creating a new measure theory where all sets are measurable has been explored through the Solovay model and other similar models. While the full Axiom of Choice is indispensable for many areas of mathematics, the Solovay model demonstrates that the existence of nonmeasurable sets is not a universal truth but rather a consequence of the choice of axioms. This opens up a rich area of research, challenging our understanding of the foundations of measure theory and the nature of the real numbers.

References

Solovay, R. M. (1970). A Model of Set Theory in Which Every Set of Reals is Lebesgue Measurable. Annals of Mathematics, 92(1), 1-56. Jech, T. (2003). Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. Fremlin, D. H. (1990). Measure Theory, Volume 4: Topological Measure Spaces. Torres Fremlin.