Conditional measures and applications

Conditional probability was once scattered across journal papers and a few chapters of broader probability texts. M. M. Rao's first edition gathered the subject into a single full-length treatment, and this second edition extends that coverage. It revisits the basics of conditioning, then moves toward abstract structural theory, updating the material with recent work and applications from the last ten years.

A central concern is the difficulty of computing conditional probabilities when the underlying space is not discrete. Rao works through many examples of these obstacles. He then applies the theory to Markov processes, martingales, potential theory, Reynolds operators, and sufficiency in statistics. Later chapters turn to noncommutative probability, showing how conditioning works in general algebraic settings, including parts of operator algebras and so-called free random variables. The book also examines existence and construction questions from the viewpoint of Bishop and Brouwer's constructive analysis.

Each chapter closes with open problems, and the text connects the subject to several other areas of mathematics. It is written for both newcomers and experienced mathematicians, and it covers conditional probability and expectation from elementary ideas through advanced abstract levels.