---
tag: ABEFWF4JHH
kind: concept
name: Standard Borel space
msc: [28A05, "03E15", 54H05]
specializes: [N4KDT83V3S]
sources:
  - url: https://en.wikipedia.org/wiki/Standard_Borel_space
same_as:
  wikidata: Q25378068
  wikipedia: Standard Borel space
---

## Summary

A standard Borel space is a measurable space whose measurable sets are the Borel sets of some Polish topology.
These are the measurable spaces met in practice, such as the real line, $[0, 1]$ and spaces of sequences.
By a theorem of Kuratowski, two standard Borel spaces are isomorphic exactly when they have the same cardinality.

## Explanations

### L1 human

Many different-looking spaces carry the same measurable structure.
The real line, the unit interval and the plane look different as topological spaces, but as measurable spaces with their Borel sets they are all the same.
A standard Borel space is a measurable space that comes from a nice topological space in this way.
Up to isomorphism there are only few of them: the finite ones, the countably infinite one and the one of the real line.