Sigma-Algebras on Spaces of Probability Measures

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.