Selected Topics of Invariant Measures in Polish Groups

Gogi Pantsulaia
Department of Mathematics, Georgian Technical University, Tbilisi, Georgia

Series: Mathematics Research Developments
BISAC: MAT029000

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This book explores a number of new applications of invariant quasi-finite diffused Borel measures in Polish groups for a solution of various problems stated by famous mathematicians (for example, Carmichael, Erdos, Fremlin, Darji and so on). By using natural Borel embeddings of an infinite-dimensional function space into the standard topological vector space of all real-valued sequences, (endowed with the Tychonoff topology) a new approach for the construction of different translation-invariant quasi-finite diffused Borel measures with suitable properties and for their applications in a solution of various partial differential equations in an entire vector space is proposed. (Imprint: Nova)

Preface

Introduction

Chapter 1. On Ordinary and Standard Lebesgue Measures in R∞

Chapter 2. On Uniformly Distributed Sequences of an Increasing Family of Finite Sets in Infinite-Dimensional Rectangles

Chapter 3. Change of Variable Formula for the α-Ordinary Lebesgue Measure in RN

Chapter 4. On Existence and Uniqueness of Generators of Shy Sets in Polish Groups

Chapter 5. On a Certain Criterion of Shyness for Subsets in the Product of Unimodular Polish Groups that are not Compact

Chapter 6. On Ordinary and Standard ”Lebesgue Measures” in Separable Banach Spaces

Chapter 7. On a Standard Product of an Arbitrary Family of σ-Finite Borel Measures with Domain in Polish Spaces

Chapter 8. On Strict Standard and Strict Ordinary Products of Measures and Some of their Applications

Chapter 9. On an Explicit Representation of a Particular Solution of the Non-Homogeneous Differential Equation of the Higher Order with Real Constant Coefficients

Chapter 10. An Invariant Measure for the Non-Homogeneous Ordinary Differential Equation of Infinite Order with Real Constant Coefficients

Chapter 11. Description of the Behaviour of von Foerster-Lasota Phase Motions in R∞ in Terms of Ordinary and Standard “Lebesgue Measures”

Chapter 12. On Uniformly Distributed Sequences on [− 1/2, 1/2]

Chapter 13. An Expansion into an Infinite-Dimensional Multiple Trigonometric Series of a Square Integrable Function in R∞

Chapter 14. On Questions of U. Darji and D. Fremlin

Chapter 15. On a Certain Modification of P. Erdös Problem for Translation-Invariant Quasi-Finite Diffused Borel Measures in Polish Groups that are not Locally Compact

Chapter 16. On a Certain Version of the Erdös Problem

Chapter 17. Appendix

References

Index

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