Standard Borel room

In measure theory , a branch of mathematics , standard Borel spaces are a very general class of measure spaces .

definition

A measurement space is a standard Borel space if it is isomorphic to a Polish space with its Borel σ-algebra . ${\displaystyle (\Omega ,{\mathcal {A}})}$

classification

Standard Borel rooms are classified by their cardinality . In particular, every uncountable standard Borel space is isomorphic to the real numbers with their Borel σ-algebra.

properties

• If there are two σ-algebras in a space for which are standard Borel spaces, then is . ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}\subset {\mathcal {B}}}$ ${\displaystyle (X,{\mathcal {A}}),(X,{\mathcal {B}})}$ ${\displaystyle {\mathcal {A}}={\mathcal {B}}}$
• For every bijective measurable image between standard Borel spaces, the reverse image can also be measured.
• A mapping between standard Borel spaces is measurable if its graph is a measurable subset of the product space.
• The completion of a standard Borel space provided with a probability measure is a standard probability space .

literature

• Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).

Web links

• Standard Borel space (Encyclopedia of Mathematics)