Math 2.0

Measure and integration Definition

Borel sigma-algebra

Builds on Sigma-algebraReal numbersInterval

Definition

Let XX be a topological space. The Borel σ-algebra ℬ(X)\mathcal{B}(X) is the smallest σ-algebra on XX that contains each open subset of XX. It is the intersection of all σ-algebras on XX that contain the open sets. A Borel set of XX is an element of ℬ(X)\mathcal{B}(X).

Remarks

On the real line, ℬ(ℝ)\mathcal{B}(\mathbb{R}) is also the σ-algebra generated by the open intervals (a,b)(a, b). The half-lines (−∞,b](-\infty, b] with b∈ℝb \in \mathbb{R} also generate it. Each open set, each closed set, each countable set and each interval of ℝ\mathbb{R} is a Borel set. Some subsets of ℝ\mathbb{R} are not Borel sets. For example, the Vitali sets are not Lebesgue measurable, so they are not Borel sets.

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptElement, definition, no evidence levelElementdefinitionIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionProposition, concept, no evidence levelPropositionconceptRational numbers, object, no evidence levelRational numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionUnion, definition, no evidence levelUniondefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementPower set, definition, no evidence levelPower setdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionComplement, definition, no evidence levelComplementdefinitionInterval, definition, no evidence levelIntervaldefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionBorel sigma-algebra, definition, no evidence levelBorel sigma-algebradefinition
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  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Rational numbers object
  7. Real numbers object
  8. Set definition
  9. Ring of integers statement
  10. Disjunction definition
  11. Implication definition
  12. Predicate definition
  13. Union definition
  14. The rationals form a field statement
  15. Universal quantifier definition
  16. Subset definition
  17. The rationals form an ordered field statement
  18. Power set definition
  19. Universal set definition
  20. Order on the reals definition
  21. Complement definition
  22. Interval definition
  23. Sigma-algebra definition
  24. Borel sigma-algebra definition

Where this leads

328 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Borel_set, Generating the Borel algebra
About this page
Tag0BVHZXWKCD
URIm2://0BVHZXWKCD
Kinddefinition
WritingW1 stated
MSC28A05
DefinesBorel sigma-algebra
UsesSigma-algebra, Real numbers, Interval

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