The nonmeasurability of Bernstein sets and related topics

TL;DR

This paper investigates Bernstein sets, proving they are nonmeasurable, lack the Baire property, and do not possess the perfect set property, thereby deepening understanding of their topological and measure-theoretic characteristics.

Contribution

It establishes that all Bernstein sets are nonmeasurable, lack the Baire property, and do not have the perfect set property, providing new insights into their foundational properties.

Findings

Bernstein sets are nonmeasurable.

Bernstein sets do not have the Baire property.

Bernstein sets lack the perfect set property.

Abstract

In this article, we shall explore the constructions of Bernstein sets, and prove that every Bernstein set is nonmeasurable and doesn't have the property of Baire. We shall also prove that Bernstein sets don't have the perfect set property.