1. Any measure defined on the Borel sets is called a Borel measure. 2. Some authors define a Haar measure on Baire sets rather than Borel sets . 3. Note that the boldface hierarchy is just the standard hierarchy of Borel sets . 4. Classical descriptive set theory includes the study of regularity properties of Borel sets . 5. Each lightface Borel set has infinitely many distinct codes. 6. Just beyond the Borel sets in complexity are the analytic sets and coanalytic sets. 7. Every Baire set is a Borel set . 8. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.9. The generalization to Borel sets is more involved, and requires the theory of Suslin sets. 10. Although any continuous preimage of a Borel set is Borel, not all analytic sets are Borel sets.
