Gaussian measures

Abstract

This chapter introduces the basic properties of Gaussian measures and gives an example of an ergodic quasi-invariant measure, which is not equivalent to any Gaussian measure; it also presents an introductory discussion of the most commonly used type of Gaussian measure—namely, Wiener measure. A criterion for the equivalence or mutual singularity of Gaussian measures is described. A very significant problem is that of determining those types of measures on a given linear measurable space that are quasi-invariant relative to a prescribed linear subspace; the investigation of equivalence and singularity of Gaussian measures constitutes a preliminary step in this direction. A class of Gaussian measure spaces that are useful in connection with harmonic analysis on linear topological spaces and some important problems in the theory of quasi-invariant measures are also described in the chapter, illustrating the situation by reference to Gaussian measures. © 1972, Elsevier B.V. All rights reserved.