category of Borel spaces

Definition 0.1.

The category of Borel spaces $\\mathbb{B}$ has, as its objects, all Borel spaces $(X_{b};\\mathcal{B}(X_{b}))$ , and as its morphisms the Borel morphisms $f_{b}$ between Borel spaces; the Borel morphism composition is defined so that it preserves the Borel structure determined by the $\\sigma$ -algebra of Borel sets.

Remark 0.1.

The category of (standard) Borel G-spaces $\\mathbb{B}_{G}$ is defined in a similar manner to $\\mathbb{B}$ , with the additional condition that Borel G-space morphisms commute withthe Borel actions $a:G\\times X\\to X$ defined as Borel functions (http://planetmath.org/BorelGroupoid)(or Borel-measurable maps). Thus, $\\mathbb{B}_{G}$ is a subcategory of $\\mathbb{B}$ ; in its turn, $\\mathbb{B}$ is a subcategory of $\\mathbb{T}op$ –the category of topological spaces and continuousfunctions.

The category of rigid Borel spaces can be defined as above with the additional condition that theonly automorphism $f:X_{b}\\to X_{b}$ (bijection) is the identity $1_{(X_{b};\\mathcal{B}(X_{b}))}$ .

单词	CategoryOfBorelSpaces
释义	category of Borel spaces Definition 0.1. The category of Borel spaces $\\mathbb{B}$ has, as its objects, all Borel spaces $(X_{b};\\mathcal{B}(X_{b}))$ , and as its morphisms the Borel morphisms $f_{b}$ between Borel spaces; the Borel morphism composition is defined so that it preserves the Borel structure determined by the $\\sigma$ -algebra of Borel sets. Remark 0.1. The category of (standard) Borel G-spaces $\\mathbb{B}_{G}$ is defined in a similar manner to $\\mathbb{B}$ , with the additional condition that Borel G-space morphisms commute withthe Borel actions $a:G\\times X\\to X$ defined as Borel functions (http://planetmath.org/BorelGroupoid)(or Borel-measurable maps). Thus, $\\mathbb{B}_{G}$ is a subcategory of $\\mathbb{B}$ ; in its turn, $\\mathbb{B}$ is a subcategory of $\\mathbb{T}op$ –the category of topological spaces and continuousfunctions. The category of rigid Borel spaces can be defined as above with the additional condition that theonly automorphism $f:X_{b}\\to X_{b}$ (bijection) is the identity $1_{(X_{b};\\mathcal{B}(X_{b}))}$ .
随便看	proof of the converse of Lagrange’s theorem for finite cyclic groups ProofOfTheCorrespondenceBetweenEven2superperfectNumbersAndMersennePrimes proof of the correspondence between even 2-superperfect numbers and Mersenne primes ProofOfTheDebutTheorem ProofOfTheDeterminantConditionForASequenceOfVectors proof of the determinant condition for a sequence of vectors ProofOfTheDimensionTheoremForSubspaces proof of the dimension theorem for subspaces proof of the début theorem ProofOfTheExistenceOfTranscendentalNumbers proof of the existence of transcendental numbers ProofOfTheFundamentalTheoremOfAlgebraLiouvillesTheorem proof of the fundamental theorem of algebra (Liouville’s theorem) ProofOfTheFundamentalTheoremOfCalculus proof of the fundamental theorem of calculus ProofOfTheJordanHolderDecompositionTheorem proof of the Jordan Hölder decomposition theorem ProofOfTheoremAboutCyclicSubspaces proof of theorem about cyclic subspaces ProofOfTheoremForNormalMatrices proof of theorem for normal matrices ProofOfTheoremOnEquivalentValuations proof of theorem on equivalent valuations ProofOfTheoremsInAdditivelyIndecomposable proof of theorems in additively indecomposable