Hyperfinite

Hyperfinite is an adjective used in several areas of mathematics to describe objects that are finite in a limiting or approximative sense. In each context, a hyperfinite object can be built as an increasing union or inductive limit of finite substructures, preserving many properties of finite systems.

In operator algebras, a von Neumann algebra M is hyperfinite if it is the von Neumann algebra generated by an increasing sequence of finite-dimensional subalgebras whose union is dense in M in the appropriate topology. Equivalently, M is approximately finite-dimensional (AFD). The separable hyperfinite II1 factor, denoted R, is the unique separable hyperfinite II1 von Neumann algebra up to isomorphism. It can be realized as the inductive limit of matrix algebras M2(C) embedded successively into larger matrix algebras. R is amenable and occupies a central role in the classification of II1 factors.

In measured group theory and descriptive set theory, hyperfinite also describes certain equivalence relations. A measured or Borel equivalence relation E is hyperfinite if it can be written as an increasing union of finite subequivalence relations. The Connes–Feldman–Weiss theorem states that every ergodic hyperfinite probability-measure-preserving II1 equivalence relation is orbit equivalent to a Z-action. In the Borel context, E0, the eventual agreement relation on sequences of zeros and ones, is a canonical example of a hyperfinite Borel equivalence relation.

In nonstandard analysis, a hyperfinite set is an internal set that is finite in the nonstandard sense (indexed by an unlimited hypernatural) but may be externally infinite, allowing finite-like arguments to approximate infinite constructions.