category of Borel spaces

Definition 0.1.

The category of Borel spaces 𝔹 has, as its objects, all Borel spacesPlanetmathPlanetmath (Xb;ℬ⁒(Xb)), and as its morphisms the Borel morphisms fb between Borel spaces; the Borel morphism composition is defined so that it preserves the Borel structure determined by the Οƒ-algebraMathworldPlanetmath of Borel sets.

Remark 0.1.

The category of (standard) Borel G-spaces 𝔹G is defined in a similar manner to 𝔹, with the additional condition that Borel G-space morphisms commute with the Borel actions a:GΓ—Xβ†’X defined as Borel functions ( (or Borel-measurable maps). Thus, 𝔹G is a subcategory of 𝔹; in its turn, 𝔹 is a subcategory of 𝕋⁒o⁒p–the category of topological spaces and continuous functionsMathworldPlanetmathPlanetmath.

The category of rigid Borel spaces can be defined as above with the additional condition that the only automorphismPlanetmathPlanetmathPlanetmath f:Xbβ†’Xb (bijection) is the identityPlanetmathPlanetmathPlanetmath 1(Xb;ℬ⁒(Xb)).

Title category of Borel spaces
Canonical name CategoryOfBorelSpaces
Date of creation 2013-03-22 18:25:01
Last modified on 2013-03-22 18:25:01
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 16
Author bci1 (20947)
Entry type Definition
Classification msc 54H05
Classification msc 28A05
Classification msc 28A12
Classification msc 28C15
Synonym category of measure spaces
Related topic Category
Related topic BorelSpace
Related topic BorelGSpace
Related topic BorelMorphism
Related topic CategoryOfPointedTopologicalSpaces
Related topic CategoryOfSets
Related topic CategoryOfPolishGroups
Related topic IndexOfCategories
Defines composition of Borel morphisms
Defines category of rigid Borel spaces