Measure theory for statisticians. Foundations of stochastics (Q493906)

This is a nice introduction to measure theory which addresses students having some basic knowledge of probability theory and a strong interest in statistics, but possibly only a limited background in mathematics. To make the subject accessible to this readership, the author recalls in great detail definitions and results from undergraduate mathematics, but not those of probability theory, which are assumed to be known. Moreover, in order to concentrate on the essential ideas of measure theory, he also avoids the most technical proofs and provides references to the literature instead. The first chapters are quite canonical and are devoted to \(\sigma\)-algebras and other systems of sets, measures and other set functions, measurable functions, and the Lebesgue integral. The following chapters concern more advanced topics like \( L^p\)-spaces, types of convergence, product measures, Fubini's theorem, the convolution of measures, and the Radon-Nikodym theorem. The final chapters on conditional expectations and likelihood functions are part of advanced probability theory and aim at applications in statistics. It is remarkable that essentially every chapter on measure theory starts with a brief motivation from probability theory. These motivations could have been confirmed if, once the corresponding part of measure theory has been developed, the circle had been closed by presenting some applications of measure theory in probability theory, which would give a reward to the reader. Throughout the book, probability measures are called normed measures, which is not beneficial with regard to the intended readership. Nevertheless, the core of this book is well-written and contains many good explanations, helpful examples and illustrations, and numerous exercises.