History

… Its Descriptive Set Theoretical Origin#

A perennial strand in set theory is the constructibility and definability of subsets of the continuum, which is precisely the subject of descriptive set theory. Descriptive set theory is actually the most classical and orthodox set theory, in that the problem it tackles with, and the way it tackles, is most similar to the concerns that originated set theory and how Cantor tackled them. (See Akihiro Kanamori's The Emergence of Descriptive Set Theory in Jaako Hintikka ed., From Dedekind to Godel,)

The original historical origin of set theory has things to do with trigonometric series. Functional analysis in fact spurred Cantor's interest in point sets and inspired his discovery of transfinite numbers, in response to Riemann's investigations of trignometric series and the related study of discontinuous functions, which led him to critique the previous attempts to define irrational numbers in terms of infinite series, and try to develop a satisfactory theory himself that do not presuppose their existence, and the iteration of the limit operation led him to the discovery of derived sets and transfinite numbers. (See Joseph M. Dauben, Georg Cantor)

Measure theory and probability theory are in close association with set theory. Gylden's problem in continued fractions was essentially about the nature of real numbers. (See Jan van Plato's Creating Modern Probability)