A Polish space (group) is a separable, completely metrizable topological space (group). This book is about actions of Polish groups, in connection with – or from the point of view of – the subject of descriptive set theory. Descriptive set theory is the study of definable sets and functions in Polish spaces. The basic classes of definable sets are the classes of Borel, analytic and co–analytic sets, and these certainly constitute the main topic of the book, but at times we also consider other classes of definable sets.

The structure of Borel actions of Polish locally compact, i.e., second countable locally compact, topological groups has long been studied in ergodic theory, operator algebras and group representation theory. See, for example, Auslander–Moore [66], Feldman–Hahn–Moore [78], Glimm [61], Kechris [92a], Mackey [57, 62, 89], Moore [82], Ramsay [82, 85], Sinai [89], Varadarajan [63], Vershik–Fedorov [87], Zimmer [84] for a sample of this work. This is closely related to the subject matter of this book. More recently, there has been increasing interest in an extension of the above: studying the structure of Borel actions of arbitrary, not necessarily locally compact, Polish groups.