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Symmetries and invariance principles play an important role in various branches of mathematics. This book deals with measures having weak symmetry properties. Even mild conditions ensure that all invariant Borel measures on a second countable locally compact space can be expressed as images of specific product measures under a fixed mapping. The results derived in this book are interesting for their own and, moreover, a number of carefully investigated examples underline and illustrate their usefulness and applicability for integration problems, stochastic simulations and statistical applications.
List of contents
Introduction, Main Theorems: Definitions and Preparatory Lemmata; Definition of Property (*) and Its Implications (Main Theorems); Supplementary Expositions and an Alternate Existence Proof.- Significance, Applicability and Advantages.- Applications: Central Definitions, Theorems and Facts; Equidistribution on the Grassmannian Manifold and Chirotopes; Conjugation-invariant Probability Measures on Compact Connected Lie Groups; Conjugation-invariant Probability Measures on SO(n); Conjugation-invariant Probability Measures on SO(3); The Theorem of Iwasawa and Invariant Measures on Lie Groups; QR-Decomposition on GL(n); Polar Decomposition on GL(n); O(n)-invariant Borel Measures on Pos(n); Biinvariant Borel Measures on GL(n); Symmetries on Finite Spaces.- References.- Glossary.- Index.
Summary
Symmetries and invariance principles play an important role in various branches of mathematics. This book deals with measures having weak symmetry properties. Even mild conditions ensure that all invariant Borel measures on a second countable locally compact space can be expressed as images of specific product measures under a fixed mapping. The results derived in this book are interesting for their own and, moreover, a number of carefully investigated examples underline and illustrate their usefulness and applicability for integration problems, stochastic simulations and statistical applications.
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From the reviews:
"This book treats measures which are invariant under group actions … . great care is given to make the book easily accessible to readers from a great variety of different backgrounds in keeping the mathematical prerequisites to a minimum and by making the exposition essentially self-contained. For a better understanding basic definitions and theorems are always illustrated by simple examples familiar from calculus." (Werner Strauß, Zentralblatt MATH, Vol. 1065, 2005)
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From the reviews:
"This book treats measures which are invariant under group actions ... . great care is given to make the book easily accessible to readers from a great variety of different backgrounds in keeping the mathematical prerequisites to a minimum and by making the exposition essentially self-contained. For a better understanding basic definitions and theorems are always illustrated by simple examples familiar from calculus." (Werner Strauß, Zentralblatt MATH, Vol. 1065, 2005)
