Borel Sets

Some articles on set, borel, borel sets, sets:

Borel Determinacy Theorem - Background - Topology
... For a given set A, whether a subset of Aω will be determined depends to some extent on its topological structure ... For the purposes of Gale–Stewart games, the set A is endowed with the discrete topology, and Aω endowed with the resulting product topology, where Aω is ... In particular, when A is the set {0,1}, the topology defined on Aω is exactly the ordinary topology on Cantor space, and when A is the set of natural numbers, it is ...
Haar Measure - Haar's Theorem
... a unique countably additive, nontrivial measure μ on the Borel subsets of G satisfying the following properties The measure μ is left-translation-invariant μ(gE) = μ(E ... The measure μ is finite on every compact set μ(K) < ∞ for all compact K The measure μ is outer regular on Borel sets E The measure μ is inner ... properties that μ(U) > 0 for every non-empty open Borel subset U ...
Descriptive Set Theory - Borel Sets - Regularity Properties of Borel Sets
... Classical descriptive set theory includes the study of regularity properties of Borel sets ... For example, all Borel sets of a Polish space have the property of Baire and the perfect set property ... Modern descriptive set theory includes the study of the ways in which these results generalize, or fail to generalize, to other classes of subsets of Polish spaces ...
Infinity-Borel Set - Incorrect Definition
... description at the top of this article as claiming that the ∞-Borel sets are the smallest class of subsets of containing all the open sets and closed under ... That is, one might wish to dispense with the ∞-Borel codes altogether and try a definition like this For each ordinal α define by transfinite recursion Bα as follows B0 is ... For a given even ordinal α, Bα+1 is the union of Bα with the set of all complements of sets in Bα ...

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