Borel Determinacy Theorem - Borel Determinacy

Borel Determinacy

Donald A. Martin (1975) proved that for any set A, all Borel subsets of Aω are determined. Because the original proof was quite complicated, Martin published a shorter proof in 1982 that did not require as much technical machinery. In his review of Martin's paper, Drake describes the second proof as "surprisingly straightforward."

The field of descriptive set theory studies properties of Polish spaces (essentially, complete separable metric spaces). The Borel determinacy theorem has been used to establish many properties of Borel subsets of these spaces. For example, all Borel subsets of Polish spaces have the perfect set property and the property of Baire.