This monograph presents a broad overview on conditioning with historical notes. It starts with principles, ideas and concepts for conditional probability measures. It then proceeds to a general concept based on conditional expectations that coincides with the Radon-Nikodým derivative. Some axiomatic definitions of conditioning are introduced. Later the conditional expectation is shown to be a particular case of a projection operator acting on functional spaces. The remaining theory, conditions and properties of conditioning are presented and discussed with several different backgrounds. The author is also interested in computational aspects of conditioning and in applications to mathematical statistics and random processes.

One chapter is devoted to sufficient statistics, which are closely related or, more correctly, defined by means of conditioning. Later chapters deals with martingales, submartingales and Markov processes. For interested readers, the last two chapters describe the impact of conditioning on modern analysis and present conditioning concepts in general spaces as some subset of functions. The latter chapters give advanced reading on the subject from a general point of view. This comprehensive monograph on conditioning can be a valuable help for anyone interested in probability theory and the connection to abstract analysis. I recommend the book to any reader interested in probability and measure theory oriented to conditioning. The book will be convenient as a textbook for introductory, tutorial and advanced courses on conditioning.

Reviewer:

pl