1. Descriptive set theory was developed by Luzin and others following the work of Borel and Lebesgue on real-valued functions. After Kleene’s investigations of hierarchies of number theoretic predicates it became clear that these hierarchies are closely related to Luzin’s hierarchy of projective sets. The introduction of recursion theoretical methods brought about new developments in descriptive set theory.
2. “Classical” descriptive set theory is treated in the books of Luzin [1930] and Kuratowski [1966]; both books have an extensive bibliography. For “modern” descriptive set theory, we refer the reader to Martin’s expository article [1977].
3. Borel sets were introduced by Borel in [1905]. Lebesgue in [1905] proved in effect Lemma 39.1. Suslin’s discovery of an error in a proof in Lebesgue article led to a construction of an analytic non-Borel set and introduction of the operation A. The basic results on analytic sets as well as Theorem 93 appeared in Suslin’s article [1917].
4. Projective sets were introduced by Luzin [1925] and [1927b], and Sierpiński [1925] and [1927]. The present notation (I and n) appeared first in the paper [1959b] of Addison who noticed the analogy between Luzin’s hierarchy of projective sets and Kleene’s hierarchy of analytical predicates [1955].
5. Lemma 39.4: Luzin [1930].