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Monotone Class Lemma
In measure theory and probability, the monotone class theorem connects monotone classes and -algebras. The theorem says that the smallest monotone class containing an algebra of sets G is precisely the smallest -algebra containing G. It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem. Definition of a monotone class A is a family (i.e. class) M of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means M has the following properties: # if A_1, A_2, \ldots \in M and A_1 \subseteq A_2 \subseteq \cdots then A_i \in M, and # if B_1, B_2, \ldots \in M and B_1 \supseteq B_2 \supseteq \cdots then B_i \in M. Monotone class theorem for sets Monotone class theorem for functions Proof The following argument originates in Rick Durrett's Probability: Theory and Examples. Results and applications As a corollary, if G is a ring of sets (The) Ring(s) may ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Families Of Sets
Family (from ) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictability, structure, and safety as members mature and learn to participate in the community. Historically, most human societies use family as the primary purpose of attachment, nurturance, and socialization. Anthropologists classify most family organizations as matrifocal (a mother and her children), patrifocal (a father and his children), conjugal (a married couple with children, also called the nuclear family), avuncular (a man, his sister, and her children), or extended (in addition to parents, spouse and children, may include grandparents, aunts, uncles, or cousins). The field of genealogy aims to trace family lineages through history. The family is also an important economic unit studied in family economics. The word "families" can be used metaphorically t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigma-ring
In mathematics, a nonempty collection of sets is called a -ring (pronounced ''sigma-ring'') if it is closed under countable union and relative complementation. Formal definition Let \mathcal be a nonempty collection of sets. Then \mathcal is a -ring if: # Closed under countable unions: \bigcup_^ A_ \in \mathcal if A_ \in \mathcal for all n \in \N # Closed under relative complementation: A \setminus B \in \mathcal if A, B \in \mathcal Properties These two properties imply: \bigcap_^ A_n \in \mathcal whenever A_1, A_2, \ldots are elements of \mathcal. This is because \bigcap_^\infty A_n = A_1 \setminus \bigcup_^\left(A_1 \setminus A_n\right). Every -ring is a δ-ring but there exist δ-rings that are not -rings. Similar concepts If the first property is weakened to closure under finite union (that is, A \cup B \in \mathcal whenever A, B \in \mathcal) but not countable union, then \mathcal is a ring but not a -ring. Uses -rings can be used instead of -fields ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Sets
(The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a Japanese horror media franchise based on the novel series by Koji Suzuki ** ''Ring'' (film), or ''The Ring'', a 1998 Japanese horror film by Hideo Nakata *** ''The Ring'' (2002 film), an American horror film, remake of the 1998 Japanese film ** ''Ring'' (1995 film), a TV film ** ''Rings'' (2005 film), a short film by Jonathan Liebesman ** ''Rings'' (2017 film), an American horror film * "Ring", a season 3 episode of ''Servant'' (TV series) Gaming * ''Ring'' (video game), 1998 * Rings (''Sonic the Hedgehog''), a collectible in ''Sonic the Hedgehog'' games Literature * ''Ring'' (Baxter novel), a 1994 science fiction novel * ''Ring'' (Alexis novel), a 2021 Canadian novel by André Alexis * ''Ring'' (novel series), a Japanese nov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin System
A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set \Omega satisfying a set of axioms weaker than those of -algebra. Dynkin systems are sometimes referred to as -systems (Dynkin himself used this term) or d-system. These set families have applications in measure theory and probability. A major application of -systems is the - theorem, see below. Definition Let \Omega be a nonempty set, and let D be a collection of subsets of \Omega (that is, D is a subset of the power set of \Omega). Then D is a Dynkin system if # \Omega \in D; # D is closed under complements of subsets in supersets: if A, B \in D and A \subseteq B, then B \setminus A \in D; # D is closed under countable increasing unions: if A_1 \subseteq A_2 \subseteq A_3 \subseteq \cdots is an increasing sequenceA sequence of sets A_1, A_2, A_3, \ldots is called if A_n \subseteq A_ for all n \geq 1. of sets in D then \bigcup_^\infty A_n \in D. It is easy to check that any Dyn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rick Durrett
Richard Timothy Durrett is an American mathematician known for his research and books on mathematical probability theory, stochastic processes and their application to mathematical ecology and population genetics. Education and career He received his BS and MS at Emory University in 1972 and 1973 and his Ph.D. at Stanford University in 1976 under advisor Donald Iglehart. From 1976 to 1985 he taught at UCLA. From 1985 until 2010 was on the faculty at Cornell University, where his students included Claudia Neuhauser. Since 2010, Durrett has been a professor at Duke University. He was elected to the United States National Academy of Sciences in 2007. In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .... Durrett is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indicator Function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator function of is the function \mathbf_A defined by \mathbf_\!(x) = 1 if x \in A, and \mathbf_\!(x) = 0 otherwise. Other common notations are and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, \mathbf_(x) = \left x\in A\ \right For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition Given an arbitrary set , the indicator function of a subset of is the function \mathbf_A \colon X \mapsto \ defined by \operatorname\mathbf_A\!( x ) = \begin 1 & \text x \in A \\ 0 & \text x \notin A \,. \end The Iverson bracket provides the equivalent notation \left x\in A\ \right/math> or that can be used instead of \mathbf_\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (mathematics)
In mathematics, a subset of a given set (mathematics), set is closed under an Operation (mathematics), operation on the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a ''collection'' of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The ''closure'' of a subset under some operations is the smallest superset that is closed under these operations. It is often called the ''span'' (for example linear span) or the ''generated set''. Definitions Let be a set (mathematics), set equipped with one or several methods for producing elements of from other elements of .Operation (mathematics), Operations and (partial function, partial) multivar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Family Of Sets
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. Additionally, a family of sets may be defined as a function from a set I, known as the index set, to F, in which case the sets of the family are indexed by members of I. In some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class. A finite family of subsets of a finite set S is also called a '' hypergraph''. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions. Examples The set of all subsets of a given set S is called the pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fubini's Theorem
In mathematical analysis, Fubini's theorem characterizes the conditions under which it is possible to compute a double integral by using an iterated integral. It was introduced by Guido Fubini in 1907. The theorem states that if a function is Lebesgue integrable on a rectangle X\times Y, then one can evaluate the double integral as an iterated integral:\, \iint\limits_ f(x,y)\,\text(x,y) = \int_X\left(\int_Y f(x,y)\,\texty\right)\textx=\int_Y\left(\int_X f(x,y) \, \textx \right) \texty. This formula is generally not true for the Riemann integral, but it is true if the function is continuous on the rectangle. In multivariable calculus, this weaker result is sometimes also called Fubini's theorem, although it was already known by Leonhard Euler. Tonelli's theorem, introduced by Leonida Tonelli in 1909, is similar but is applied to a non-negative measurable function rather than to an integrable function over its domain. The Fubini and Tonelli theorems are usually combined and for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
