At the nexus of probability theory, geometry and statistics, a Gaussian
measure is constructed on a Hilbert space in two ways: as a product
measure and via a characteristic functional based on Minlos-Sazonov
theorem. As such, it can be utilized for obtaining results for
topological vector spaces. Gaussian Measures contains the proof for
Fernique�s theorem and its relation to exponential moments in Banach
space. Furthermore, the fundamental Feldman-Hájek dichotomy for Gaussian
measures in Hilbert space is investigated. Applications in statistics
are also outlined. In addition to chapters devoted to measure theory,
this book highlights problems related to Gaussian measures in Hilbert
and Banach spaces. Borel probability measures are also addressed, with
properties of characteristic functionals examined and a proof given
based on the classical Banach-Steinhaus theorem. Gaussian Measures is
suitable for graduate students, plus advanced undergraduate students in
mathematics and statistics. It is also of interest to students in
related fields from other disciplines. Results are presented as lemmas,
theorems and corollaries, while all statements are proven. Each
subsection ends with teaching problems, and a separate chapter contains
detailed solutions to all the problems. With its student-tested
approach, this book is a superb introduction to the theory of Gaussian
measures on infinite-dimensional spaces.
