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Hyperfinite Equivalence Relation
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descriptive Set Theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic. Polish spaces Descriptive set theory begins with the study of Polish spaces and their Borel sets. A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line \mathbb, the Baire space \mathcal, the Cantor space \mathcal, and the Hilbert cube I^. Universality properties The class of Polish spaces has several universality properties, which show that there is no loss of generality in considering Polish spaces of certain restricted for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets, there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, a set arising from choosing elements arbitrarily can be made without invoking the axiom of choice; this is, in particular, the case if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available – some distinguis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events; for example, the value assigned to "1 or 2" in a throw of a dice should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a function \mu to be a probability measure on a probability space are that: * \mu must return results in the unit interval , 1 returning 0 for the empty set and 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benjamin Weiss
Benjamin Weiss ( he, בנימין ווייס; born 1941) is an American-Israeli mathematician known for his contributions to ergodic theory, topological dynamics, probability theory, game theory, and descriptive set theory. Biography Benjamin ("Benjy") Weiss was born in New York City. In 1962 he received B.A. from Yeshiva University and M.A. from the Graduate School of Science, Yeshiva University. In 1965, he received his Ph.D. from Princeton under the supervision of William Feller. Academic career Between 1965 and 1967, Weiss worked at the IBM Research. In 1967, he joined the faculty of the Hebrew University of Jerusalem; and since 1990 occupied the Miriam and Julius Vinik Chair in Mathematics (Emeritus since 2009). Weiss held visiting positions at Stanford, MSRI, and IBM Research Center. Weiss published over 180 papers in ergodic theory, topological dynamics, orbit equivalence, probability, information theory, game theory, descriptive set theory; with notable con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amenable Group
In mathematics, an amenable group is a locally compact topological group ''G'' carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure (or mean) on subsets of ''G'', was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "''mean''". The amenability property has a large number of equivalent formulations. In the field of analysis, the definition is in terms of linear functionals. An intuitive way to understand this version is that the support of the regular representation is the whole space of irreducible representations. In discrete group theory, where ''G'' has the discrete topology, a simpler definition is used. In this setting, a group is amenable if one can say what ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meagre Set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms. The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre. Meagre sets play an important role in the formulation of the notion of Baire space and of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis. Definitions Throughout, X will be a topological space. A subset of X is called X, a of X, or of the in X if it is a countable union of nowhere dense subsets of X (where a nowhere dense set is a set whose closure has empty interior). The qualifier "in X" can be omitted if the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Growth Rate (group Theory)
In the mathematical subject of geometric group theory, the growth rate of a group with respect to a symmetric generating set describes how fast a group grows. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length ''n''. Definition Suppose ''G'' is a finitely generated group; and ''T'' is a finite ''symmetric'' set of generators (symmetric means that if x \in T then x^ \in T ). Any element x \in G can be expressed as a word in the ''T''-alphabet : x = a_1 \cdot a_2 \cdots a_k \text a_i\in T. Consider the subset of all elements of ''G'' that can be expressed by such a word of length ≤ ''n'' :B_n(G,T) = \. This set is just the closed ball of radius ''n'' in the word metric ''d'' on ''G'' with respect to the generating set ''T'': :B_n(G,T) = \. More geometrically, B_n(G,T) is the set of vertices in the Cayley graph with respect to ''T'' that are within dista ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely-generated Group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of such elements. By definition, every finite group is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated. Examples * Every quotient of a finitely generated group ''G'' is finitely generated; the quotient group is generated by the images of the generators of ''G'' under the canonical projection. * A subgroup of a finitely generated group need not be finitely generated. * A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
