Descriptive set theory and definable proper forcing are two areas
of set theory that developed quite independently of each other.
This monograph unites them and explores the connections between
them. Forcing is presented in terms of quotient algebras of various
natural sigma-ideals on Polish spaces, and forcing properties in
terms of Fubini-style properties or in terms of determined infinite
games on Boolean algebras. Many examples of forcing notions appear,
some newly isolated from measure theory, dynamical systems, and
other fields. The descriptive set theoretic analysis of operations
on forcings opens the door to applications of the theory:
absoluteness theorems for certain classical forcing extensions,
duality theorems, and preservation theorems for the countable
support iteration. Containing original research, this text
highlights the connections that forcing makes with other areas of
mathematics, and is essential reading for academic researchers and
graduate students in set theory, abstract analysis and measure
theory.
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