Sigma-Algebras on Spaces of Probability Measures
研究了度量空间上所有概率测度集合上的三种西格玛代数及其关系，给出了参数族可测性和博雷尔同构的条件，并证明了绝对连续概率测度集合的可测性。
This study is of interest as it relates to the choice of sigma-algebras to be used in non-parametric decision theoretic problems. Three sigma-algebras on the set of all probability measures on a metric space are considered, and the relationships among these are explored. General conditions, which ensure measurability of well-behaved parametric families in two of these sigma-algebras, are derived. Necessary conditions are given for a parametric family, with the Borel structure it inherits from the parameter space, to be Borel isomorphic to its image in two of these sigma-algebras. It is also shown that the set of all probability measures on a complete, separable metric space, which are absolutely continuous with respect to a given positive measure on the space, is measurable in these two sigma-algebras, and that the mapping from probability densities in Y, to the set of absolutely continuous probability measures is a Borel isomorphism.