In descriptive set theory and related areas of mathematics, a hyperfinite equivalence relation on a standard Borel space X is a Borel equivalence relation E with countable classes, that can, in a certain sense, be approximated by Borel equivalence relations that have finite classes.
Definitions Definition 1. Let X be a standard Borel space, that is; it is a measurable space which arises by equipping a Polish space X with its σ-algebra of Borel subsets (and forgetting the topology). Let E be an equivalence relation on X. We will say that E is Borel if E is a Borel subset of the cartesian product of X with itself, when equipped with the product σ-algebra. We will say that E is finite (respectively, countable) if E has finite (respectively, countable) classes. The above names might be misleading, since if X is an uncountable standard Borel space, the equivalence relation will be uncountable when considered as a set of ordered pairs from X. Definition 2. Let E be a countable Borel equivalence relation on a standard Borel space X. We will say that E is hyperfinite if E = ⋃ n ∈ N F n {\displaystyle E=\bigcup _{n\in \mathbb {N} }F_{n}} , where F n {\displaystyle F_{n}} is an increasing sequence of finite Borel equivalence relations on X. Intuitively, this means that there is a sequence finite equivalence relations on X, each finer than its predecessors, approximating E arbitrarily well.
Discussion A major area of research in descriptive set theory is the classification of Borel equivalence relations, and in particular those which are countable. Among these, finite equivalence relations are considered to be the simplest (for instance, they admit Borel transversals). Therefore, it is natural to ask whether certain equivalence relations, which are not necessarily finite, can be approximated by finite equivalence relations. This turns out to be a notion which is both rich enough to encapsulate many natural equivalence relations appearing in mathematics, yet restrictive enough to allow deep theorems to develop. It is also worthwhile to note that any countable equivalence relation E can be written down as an increasing union of finite equivalence relations. This can be done, for instance, by taking a partition of every class into classes of size two, then joining two classes in the new equivalence relation which are within the same E-class to form a partition with classes of size four, and so forth. The key observation is that this process requires the axiom of choice in general, and therefore it is not clear that this process generates Borel approximations. Indeed, there are countable Borel equivalence relations that are not hyperfinite, and so in particular the process described above will fail to generate Borel equivalence relations approximating the larger equivalence relation.
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