A general setting for the pointwise investigation of determinacy
Yurii Khomskii
Abstract:
It is well-known that if we assume a large class of sets of reals to
be determined then we may conclude that all sets in this class have
certain regularity properties: we say that determinacy implies
regularity properties "classwise". In [Lo05] the "pointwise" relation
between determinacy and certain regularity properties (namely the
Marczewski-Burstin algebra of arboreal forcing notions and a
corresponding weak version) was examined.
An open question was how this result extends to topological forcing
notions whose natural measurability algebra is the class of sets
having the Baire property. We study the relationship between the two
cases, and using a definition which adequately generalizes both the
Marczewski-Burstin algebra of measurability and the Baire property,
prove results similar to [Lo05].
We also show how this can be further generalized for the purpose of
comparing algebras of measurability of various forcing notions.
References:
[Lo05] Benedikt Loewe, "The pointwise view of determinacy: arboreal
forcings, measurability, and weak measurability" Rocky Mountain
Journal of Mathematics 35, 1233-1249 (2005).
Keywords: Set theory; Determinacy; Regularity properties