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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 a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
