Illinois Journal of Mathematics
- Illinois J. Math.
- Volume 27, Issue 4 (1983), 624-658.
Singularity and absolute continuity with respect to strategic measures
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.
Illinois J. Math., Volume 27, Issue 4 (1983), 624-658.
First available in Project Euclid: 20 October 2009
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 28A35: Measures and integrals in product spaces
Secondary: 03E55: Large cardinals 60G30: Continuity and singularity of induced measures
Armstrong, Thomas E.; Prikry, Karel. Singularity and absolute continuity with respect to strategic measures. Illinois J. Math. 27 (1983), no. 4, 624--658. doi:10.1215/ijm/1256046253. https://projecteuclid.org/euclid.ijm/1256046253