This book aims to put strong reasonable mathematical senses in notions of objectivity and subjectivity for consistent estimations in a Polish group by using the concept of Haar null sets in the corresponding group. This new approach - naturally dividing the class of all consistent estimates of an unknown parameter in a Polish group into disjoint classes of subjective and objective estimates - helps the reader to clarify some conjectures arising in the criticism of null hypothesis significance testing. The book also acquaints readers with the theory of infinite-dimensional Monte Carlo integration recently developed for estimation of the value of infinite-dimensional Riemann integrals over infinite-dimensional rectangles. The book is addressed both to graduate students and to researchers active in the fields of analysis, measure theory, and mathematical statistics.

This book aims to put strong reasonable mathematical senses in notions of objectivity and subjectivity for consistent estimations in a Polish group by using the concept of Haar null sets in the corresponding group. This new approach - naturally dividing the class of all consistent estimates of an unknown parameter in a Polish group into disjoint classes of subjective and objective estimates - helps the reader to clarify some conjectures arising in the criticism of null hypothesis significance testing. The book also acquaints readers with the theory of infinite-dimensional Monte Carlo integration recently developed for estimation of the value of infinite-dimensional Riemann integrals over infinite-dimensional rectangles. The book is addressed both to graduate students and to researchers active in the fields of analysis, measure theory, and mathematical statistics.

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.

This monograph deals with certain aspects of the general theory of systems. The author develops the ergodic theory, which is the theory of quaslinvariant and invariant measures in such infinite-dimensional vector spaces which appear as models of various (physical, economic, genetic, linguistic, social, etc.) processes. The methods of ergodic theory are successful as applied to study properties of such systems. A foundation for ergodic theory was stimulated by the necessity of a consideration of statistic mechanic problems and was directly connected with the works of G. Birkhoff, Kryloff and Bogoliuboff, E. Hoph and other famous mathematicians.

This book explores a new concept of T-shy sets in Radon metric groups which is an extension of the concepts of null sets introduced by J.R. Christensen, J. Mycielski and B R Hunt, T Sauer and J A Yorke for Polish groups. This book considers various interesting applications of the theory of infinite-dimensional cellular matrices for a solution of various initial condition problems. It also includes several direct generalizations of well-known classical results (Lebesgue theorem, Weyl theorem, Rademacher theorem, etc.,) in terms of infinite-dimensional "Lebesgue measures".