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Pi System
In mathematics, a -system (or pi-system) on a set \Omega is a collection P of certain subsets of \Omega, such that * P is non-empty. * If A, B \in P then A \cap B \in P. That is, P is a non-empty family of subsets of \Omega that is closed under non-empty finite intersections.The nullary (0-ary) intersection of subsets of \Omega is by convention equal to \Omega, which is not required to be an element of a -system. The importance of -systems arises from the fact that if two probability measures agree on a -system, then they agree on the -algebra generated by that -system. Moreover, if other properties, such as equality of integrals, hold for the -system, then they hold for the generated -algebra as well. This is the case whenever the collection of subsets for which the property holds is a -system. -systems are also useful for checking independence of random variables. This is desirable because in practice, -systems are often simpler to work with than -algebras. For example, it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filter (set Theory)
In mathematics, a filter on a set X is a Family of sets, family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A,B\subset X,A\in \mathcal, and A\subset B, then B\in \mathcal A filter on a set may be thought of as representing a "collection of large subsets". Filters appear in Order theory, order, model theory, set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal (set theory), ideal. Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book ''Topologie Générale'' as an alternative to the related notion of a Net (topology), net developed in 1922 by E. H. Moore and Herman L. Smith. Filter (mathematics), Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specificall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
