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Borel space (Definition)
Definition 0.1   A Borel space $ (X; \mathcal{B}(X))$ is defined as a topological space $ X$, together with a $ \sigma$-algebra $ \mathcal{B}(X)$ of subsets of $ X$, called Borel sets.

Note A subspace of a Borel space $ (X; \mathcal{B} (X))$ is a subset $ S \subset X$ endowed with the relative Borel structure, that is the $ \sigma$-algebra of all subsets of $ S$ of the form $ S \bigcap E$, where $ E$ is a Borel subset of $ X$.

Bibliography

1
M.R. Buneci. 2006., Groupoid C*-Algebras., Surveys in Mathematics and its Applications, Volume 1: 71-98.
2
A. Connes.1979. Sur la théorie noncommutative de l' integration, Lecture Notes in Math., Springer-Verlag, Berlin, 725: 19-14.



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See Also: Borel set, $\sigma$-algebra, measurable space, Borel measure, $\sigma$--finite Borel measure and related Borel concepts, Borel groupoid, Borel morphism, $\sigma$--finite Borel measure and related Borel concepts, category of Borel spaces

Other names:  measurable space
Also defines:  Borel subset space
Keywords:  Borel space, Borel set, sigma algebra

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$\sigma$--finite Borel measure and related Borel concepts (Topic) by bci1
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Cross-references: Borel subset, Borel structure, subspace, Borel sets, subsets, topological space
There are 26 references to this entry.

This is version 9 of Borel space, born on 2008-09-15, modified 2008-11-11.
Object id is 11026, canonical name is BorelSpace.
Accessed 665 times total.

Classification:
AMS MSC: 28A05 (Measure and integration :: Classical measure theory :: Classes of sets , measurable sets, Suslin sets, analytic sets)
 54H05 (General topology :: Connections with other structures, applications :: Descriptive set theory )
 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)
 28C15 (Measure and integration :: Set functions and measures on spaces with additional structure :: Set functions and measures on topological spaces )
 60A10 (Probability theory and stochastic processes :: Foundations of probability theory :: Probabilistic measure theory)
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