There are at least two ways to develop probability theory. The more familiar path is to treat it as its own discipline, and work from intuitive examples such as coin flips and conundrums such as the Monty Hall problem. An alternative is to first develop measure theory and analysis, and then add interpretation. Bhattacharya and Waymire take the second path. To illustrate the authors' frame of reference, consider the two definitions they give of conditional expectation. The first is as a projection of L2 spaces. The authors rely on the reader to be familiar with Hilbert space operators and at a glance, the connection to probability may not be not apparent. Subsequently, there is a discusssion of Bayes's rule and other relevant probabilistic concepts that lead to a definition of conditional expectation as an adjustment of random outcomes from a finer to a coarser information set.
The main goal of Volume 4 is to develop the functional analysis setup for the universe of generalized functions. The main notion introduced in this volume is the notion of rigged Hilbert space (also known as the equipped Hilbert space, or Gelfand triple). Such space is, in fact, a triple of topological vector spaces E⊂H⊂E′, where H is a Hilbert space, E′ is dual to E, and inclusions E⊂H and H⊂E′ are nuclear operators. The book is devoted to various applications of this notion, such as the theory of positive definite generalized functions, the theory of generalized stochastic processes, and the study of measures on linear topological spaces.