Applications of Measure Theory to Statistics 1 Calculation of Improper Integrals by Using Uniformly Distributed Sequences 2 Infinite Dimensional Monte Carlo Integration 3 On structure of all real valued sequences uniformly distributed in [ 1 2;1 2] from the point of view of shyness 4 On Moore Yamasaki Kharazishvili type measures and the infinite powers of Borel diffused probability measures on R 5 On objective and strong objective consistent estimates of unknown parameters for statistical structures in a Polish group admitting an invariant metric 6 Why Null Hypothesis is rejected for “almost every” infinite sample by the Hypothesis Testing of a maximal reliability. 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.