---
tag: D9PDDPC61K
kind: definition
name: Standard Borel space
msc: [28A05, "03E15", 54H05]
defines: ABEFWF4JHH
uses: [N4KDT83V3S, 0CQHFPGJ45, 0Z2P41SVDP, 0BY52JW6NK, 0WQXY32Q8D]
sources:
  - url: https://en.wikipedia.org/wiki/Standard_Borel_space
---

## Definition

A *Polish space* is a [[topological space|0CQHFPGJ45]] $(X, \tau)$ that is [[separable|0Z2P41SVDP]] and has the following property.
There is a metric $d$ on $X$ that induces $\tau$ such that $(X, d)$ is a [[complete metric space|0BY52JW6NK]].
Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]].
It is a *standard Borel space* if $X$ has a topology $\tau$ with these two properties.
The space $(X, \tau)$ is a Polish space, and $\Sigma$ is the [[Borel σ-algebra|0WQXY32Q8D]] of $\tau$.

## Remarks

The real line $\R$ with its Borel σ-algebra is a standard Borel space, since the usual metric on $\R$ is complete and $\Q$ is a countable dense subset.
A countable set with its power set is a standard Borel space, using the discrete metric.
Let $\Sigma$ be the family of subsets $A \subseteq \R$ such that $A$ or $\R \setminus A$ is countable.
Then $(\R, \Sigma)$ 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 $\Sigma$ 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 $\R$, and that two standard Borel spaces of the same cardinality are isomorphic.