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Borel G-space
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(Definition)
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A (standard) Borel G-space is defined in connection with a standard Borel space which shall be specified first.
- a. Standard Borel space
Definition 0.1 A standard Borel space is defined as a measurable space, that is, a set $X$ equipped with a $\sigma$ -algebra $\mathcal{S}$ , such that there exists a Polish topology on $X$ with $S$ its $\sigma$ -algebra of Borel sets.
- b. Borel G-space
Definition 0.2 Let $G$ be a Polish group and $X$ a (standard) Borel space. An action $a$ of $G$ on $X$ is defined to be a Borel action if $a: G \times X \to X$ is a Borel-measurable map or a Borel function. In this case, a standard Borel space $X$ that is acted upon by a Polish group with a Borel action is called a
(standard) Borel G-space.
- c. Borel morphisms
Remark 0.1 Borel G-spaces have the nice property that the product and sum of a countable sequence of Borel G-spaces $(X_n)_{n \in N}$ are also Borel G-spaces. Furthermore, the subspace of a Borel G-space determined by an invariant Borel set is also a Borel G-space.
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"Borel G-space" is owned by bci1.
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See Also: Borel space, Borel measure, Borel groupoid, category of Borel spaces
| Also defines: |
Borel action, Borel-measurable map, standard Borel space |
| Keywords: |
Borel action, Borel-measurable map, standard Borel space, Borel function, Borel space, Polish group acting on a Borel space |
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Cross-references: invariant, subspace, sequence, countable, sum, product, property, G-spaces, isomorphisms, embeddings, homomorphisms, Borel morphisms, action, Borel space, Polish group, Borel sets, topology, measurable space, connection
There are 2 references to this entry.
This is version 11 of Borel G-space, born on 2008-09-21, modified 2009-05-28.
Object id is 11061, canonical name is BorelGSpace.
Accessed 1506 times total.
Classification:
| AMS MSC: | 22A10 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Analysis on general topological groups) | | | 22A05 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Structure of general topological groups) | | | 54H05 (General topology :: Connections with other structures, applications :: Descriptive set theory ) | | | 22A22 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Topological groupoids ) | | | 22A25 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Representations of general topological groups and semigroups) | | | 22A15 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Structure of topological semigroups) |
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Pending Errata and Addenda
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