Singularity and absolute continuity with respect to strategic measures

Abstract

Extending the result of Prikry and Sudderth that a reverse strategic product measure on $N \times N$ with diffuse marginal measures is singular to all strategic measures (i.e. purely non-strategic) we show in Section 1 that any reverse strategic product measure an $X \times Y$ (where $X$ and $Y$ are arbitrary sets) is purely non-strategic if it has purely finitely additive marginal measures. If there are no real-valued measurable cardinals so all countably additive measures are discrete the Converse is true. In Section 2, we introduce the language of split faces of probability measures as a convenient tool for discussing decompositions of probability measures. In this section we characterize which nearly strategic measures are absolutely continuous with respect to a given strategic measure. In Section 3, atomicity and non-atomicity of strategic measures are characterized. In Section 4, we deal with $\kapa$-additivity of strategic measures for an infinite cardinal $\kapa$. In Section 5, $\kapa$-uniformity of strategic measures is discussed. In Section 6, we give examples of reverse strategic product measures with diffuse marginals, one of which is countably additive, which are strategic. We also examine when a reverse strategic product measure with diffuse marginals, one of which is countably additive, may be purely non-strategic.