We survey the theory of treeable countable Borel equivalence relations.

Chapter 8 provides an introduction to Gaussian measures on a Banach space from the point of view that originated in the work of N. Wiener and was further developed by L. Gross and I. Segal. The underlying idea is that, even though it cannot fit there, the measure would like to live on the Hilbert space (the Cameron–Martin space) for which it would be the standard Gauss measure, and it is in that Hilbert space that its properties are encoded. A good deal of functional analysis is required to carry out this program, and the estimate that makes the program possible is X. Fernique’s remarkable exponential estimate. Included are derivations of M. Schilder’s large deviations theorem for Brownian motion and V. Strassen’s function space version of the law of the iterated logarithm, both of which confirm the importance of the Cameron–Martin space.

We discuss results demonstrating the complex structure of the Borel reducibility order.

We study the poset of bireducibility types of countable Borel equivalence relations and its relation to the theory of cardinal algebras.

In Chapter 4 I construct the Lévy processes (a.k.a. independent increment processes) corresponding to infinitely divisible laws. Section 4.1 provides the requisite information about the pathspace D(ℝN) of right-continuous paths with left limits, and §4.2 gives the construction of Lévy processes with discontinuous paths, the ones corresponding to infinitely divisible laws having no Gaussian part. Finally, in §4.3 I construct Brownian motion, the Lévy process with continuous paths, following the prescription given by Lévy. This section also contains a derivation of Kolmogorov’s continuity criterion for general Banach space-valued stochastic processes.

We introduce a universal space for actions and equivalence relations via spaces of subshifts.

We survey the theory of free countable Borel equivalence relations.

The intimate connection between Brownian motion of classical potential theory is described in Chapter 11. The first topic is again the representation of solutions to the Dirichlet problem in terms of the exit distribution of Brownian paths from a region. In particular, it is shown that, with probability 1, Brownian paths exit through regular points. This is followed by a discussion of the Poisson problem and its relationship, depending on dimension, to the transience or recurrence of Brownian paths. Among other things, a proof is given of F. Riesz’s representation theorem for superharmonic functions, and this result is used to introduce the concept of capacity. K. L. Chung’s formula for the capacitory potential in term of the last exit distribution of Brownian paths is derived and used to prove Wiener’s test for regularity in terms of capacity. Finally, the chapter concludes with two interesting connections, one made by F. Spitzer and the other by G. Hunt, between Brownian paths and capacity.

We survey the theory of amenable countable Borel equivalence relations.

We introduce essentially countable and reducible-to-countable Borel equivalence relations and discuss situations where they appear in various contexts.

We survey the role of structurability in the theory of countable Borel equivalence relations.

We introduce countable Borel equivalence relations and survey some of the fundamental structural results.

We introduce and study topological realizations of countable Borel equivalence relations.

We start the study of the structure of the Borel reducibility order on countable Borel equivalence relations.