These notes give details on the probability concepts of “infinitely often” and “with probability 1.” This is useful for understanding the Borel-Cantelli lemma and the strong law of large numbers. I. SEQUENCES OF EVENTS A. Events Consider a general sample space S. Recall that an event is a subset of S that has a well defined probability. That is, a set A is an event if and only if A ⊆ S and P [A...

The reader is referred to Chapter 1.6 of Wellner’s Torgnon notes, Chapter ??? of VDVW and Chapter 8.3 of Kosorok. First, a theorem using bracketing entropy. Let (F , ‖ ‖) be a subset of a normed space of real functions f : X → R. Given real functions l and u on X (but not necessarily in F), the bracket [l, u] is defined as the set of all functions f ∈ F satisfying l ≤ f ≤ u. The functions l, u ...

In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal for the study of analytic equivalence relations, we study countable Borel qua...