Welcome

This book reports on the state of a research program that I initiated in 1999. It connects the practice of proper forcing introduced by Shelah [64] with the study of various σ-ideals on Polish spaces from the point of view of abstract analysis, descriptive set theory, measure theory, etc. It turns out that the connection is far richer than I dared to imagine in the beginning. Its benefits include theorems about methodology of forcing as well as isolation of new concepts in measure theory or abstract analysis. It is my sincere hope that this presentation will help to draw attention from experts from these fields and to bring set theory and forcing closer to the more traditional parts of mathematics.

The book uses several theorems and proofs from my earlier papers; in several cases I coauthored these papers with others. The first treatment of the subject in [83] is superseded here on many accounts, but several basic theorems and proofs remain unchanged. The papers [18], [67], [82], [86], and [87] are incorporated into the text, in all cases reorganized and with significant improvements.

Many mathematicians helped to make this book what it is. Thanks should go in the first place to Bohuslav Balcar for his patient listening and enlightening perspective of the subject.