Quasi-Invariance of Product Measures Under Lie Group Perturbations: Fisher Information and $L^2$-Differentiability

Abstract

A sequence of measures on a topological space is perturbed by a sequence of elements of a Lie group acting on that space. Criteria are given for the singularity and equivalence of the corresponding product measures. These criteria extend the results of Shepp and Steele. In particular, Fisher information comes into the scene and its role is further clarified.