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Invariant Sigma-algebra
In mathematics, especially in probability theory and ergodic theory, the invariant sigma-algebra is a sigma-algebra formed by sets which are invariant set, invariant under a group action or dynamical system. It can be interpreted as of being "indifferent" to the dynamics. The invariant sigma-algebra appears in the study of ergodic systems, as well as in theorems of probability theory such as de Finetti's theorem and the Hewitt-Savage zero-one law, Hewitt-Savage law. Definition Strictly invariant sets Let (X,\mathcal) be a measurable space, and let T:(X,\mathcal)\to(X,\mathcal) be a measurable function. A measurable subset S\in \mathcal is called invariant set, invariant if and only if T^(S)=S. Equivalently, if for every x\in X, we have that x\in S if and only if T(x)\in S. More generally, let M be a group (mathematics), group or a monoid, let \alpha:M\times X\to X be a monoid action, and denote the action of m\in M on X by \alpha_m:X\to X. A subset S\subseteq X is \alpha-i ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
