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Measure and integration Definition

Standard Borel space

Builds on Measurable spaceTopological spaceSeparable spaceComplete metric spaceBorel sigma-algebra

Definition

A Polish space is a topological space (X,τ)(X, \tau) that is separable and has the following property. There is a metric dd on XX that induces τ\tau such that (X,d)(X, d) is a complete metric space. Let (X,Σ)(X, \Sigma) be a measurable space. It is a standard Borel space if XX has a topology τ\tau with these two properties. The space (X,τ)(X, \tau) is a Polish space, and Σ\Sigma is the Borel σ-algebra of τ\tau.

Remarks

The real line ℝ\R with its Borel σ-algebra is a standard Borel space, since the usual metric on ℝ\R is complete and ℚ\Q is a countable dense subset. A countable set with its power set is a standard Borel space, using the discrete metric. Let Σ\Sigma be the family of subsets A⊆ℝA \subseteq \R such that AA or ℝ∖A\R \setminus A is countable. Then (ℝ,Σ)(\R, \Sigma) is a measurable space but not a standard Borel space. Indeed, the Borel σ-algebra of a Polish space is generated by countably many sets. These are the open balls with rational radius around the points of a countable dense set. The σ-algebra Σ\Sigma is not generated by countably many sets. Equivalently, a standard Borel space is a measurable space isomorphic to a Polish space with its Borel σ-algebra. Kuratowski's theorem says that a standard Borel space is countable or has the cardinality of ℝ\R, and that two standard Borel spaces of the same cardinality are isomorphic.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptComplete metric space, concept, no evidence levelComplete metric spaceconceptMeasurable space, concept, no evidence levelMeasurable spaceconceptSeparable space, concept, no evidence levelSeparable spaceconceptTopological space, concept, no evidence levelTopological spaceconceptStandard Borel space, definition, no evidence levelStandard Borel spacedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Complete metric space concept
  3. Measurable space concept
  4. Separable space concept
  5. Topological space concept
  6. Standard Borel space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Standard_Borel_space
About this page
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Kinddefinition
WritingW0 stub
MSC28A05, 03E15, 54H05
DefinesStandard Borel space
UsesMeasurable space, Topological space, Separable space, Complete metric space, Borel sigma-algebra
Changed2 November 2026

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