Borel isomorphism

As Borel isomorphism is a relationship between two measurement areas in the measure theory , a branch of mathematics called. If two measurement spaces are Borel isomorphic, then they are the same from the point of view of dimension theory. This allows arguments and structures to be transferred from one room to the other.

definition

Two measurement spaces are given , with the corresponding Borel σ-algebra being selected as the σ-algebra . ${\displaystyle (S,{\mathcal {B}}(S)),(T,{\mathcal {B}}(T))}$

Then the two measuring spaces are called Borel isomorphic if there is a function

${\displaystyle f\colon (S,{\mathcal {B}}(S))\to (T,{\mathcal {B}}(T))}$

that has the following properties:

• ${\displaystyle f}$is bijective
• ${\displaystyle f}$is measurable

A function is called bimeasurable if both and the inverse function are measurable . ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f^{-1}}$

Borel rooms

The so-called Borel spaces are an important example of Borel isomorphism. These are measurement spaces that are Borel-isomorphic to a Borel-measurable subset of the real numbers (provided with the corresponding trace algebra ${\displaystyle \sigma }$ of Borel's σ-algebra ). ${\displaystyle \mathbb {R} }$

supporting documents

• Olav Kallenberg: Random Measures, Theory and Applications . Springer, Switzerland 2017, doi : 10.1007 / 978-3-319-41598-7 .
• AG El'kin: Borel isomorphism . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).