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related areas of mathematics, a hyperfinite equivalence relation on a standard Borel space X is a Borel equivalence relation E with countable classes, that
Hyperfinite equivalence relation
Hyperfinite_equivalence_relation
Mathematical concept for comparing objects
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments
Equivalence_relation
standard Borel spaces X and Y are Borel-isomorphic iff |X| = |Y|. Hyperfinite equivalence relation Wadge hierarchy Entourage (topology) – Topological space with
Borel_equivalence_relation
Descriptive set theory relation
encapsulates various more specific concepts, such as that of a hyperfinite equivalence relation, but is of interest in and of itself. A main area of study
Countable_Borel_relation
Calculus using a logically rigorous notion of infinitesimal numbers
R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} by the resulting equivalence relation is a hyperreal field ∗ R {\displaystyle ^{*}\mathbb {R} } , a situation
Nonstandard_analysis
*-algebra of bounded operators on a Hilbert space
are the hyperfinite type II1 factor and the hyperfinite type II∞ factor, found by Murray & von Neumann (1936). These are the unique hyperfinite factors
Von_Neumann_algebra
continuous geometry other than projective space was the projections of the hyperfinite type II factor. Menger and Birkhoff gave axioms for projective geometry
Continuous_geometry
Generalization of the real numbers
to the order relation ≤ given by the comparison rule below. The numeric forms are placed in equivalence classes; each such equivalence class is a surreal
Surreal_number
C*-algebra
counterpart of simple AF C*-algebras in the von Neumann algebra world are the hyperfinite factors, which were classified by Connes and Haagerup. In the context
Approximately finite-dimensional C*-algebra
Approximately_finite-dimensional_C*-algebra
Concept in model theory
true of hyperreal numbers. The transfer principle concerns the logical relation between the properties of the real numbers R, and the properties of a larger
Transfer_principle
Element of a nonstandard model of the reals, which can be infinite or infinitesimal
Surprisingly enough, there is a consistent way to do it. As a result, the equivalence classes of sequences that differ by some sequence declared zero will
Hyperreal_number
Type of set in mathematical logic
Relative to the ultrapower construction of the hyperreal numbers as equivalence classes of sequences ⟨ u n ⟩ {\displaystyle \langle u_{n}\rangle } of
Internal_set
Real numbers adjoined with a nil-squaring element
A relation is defined on B as follows: (a, b) ~ (c, d) when there is a u in U such that ua = c and ub = d. This relation is in fact an equivalence relation
Dual_number
Proof technique in nonstandard analysis
unlimited (infinite) element of *N. These facts can be used to prove the equivalence of the following two conditions for an internal hyperreal-valued function
Overspill
Hungarian and American mathematician and physicist (1903–1957)
continuous geometry other than projective space was the projections of the hyperfinite type II factor. In more pure lattice theoretical work, he solved the
John_von_Neumann
Extremely small quantity in calculus; thing so small that there is no way to measure it
null sequence becomes an infinitesimal in the sense of an equivalence class modulo a relation defined in terms of a suitable ultrafilter. The article by
Infinitesimal
HYPERFINITE EQUIVALENCE-RELATION
HYPERFINITE EQUIVALENCE-RELATION
HYPERFINITE EQUIVALENCE-RELATION
HYPERFINITE EQUIVALENCE-RELATION
HYPERFINITE EQUIVALENCE-RELATION
HYPERFINITE EQUIVALENCE-RELATION
HYPERFINITE EQUIVALENCE-RELATION
HYPERFINITE EQUIVALENCE-RELATION
HYPERFINITE EQUIVALENCE-RELATION