Separable σ-algebra

In measure theory , a σ-algebra is called separable or countably generated if it can be generated from a countable number of sets.

The separability of a σ-algebra plays a role in the question of when a -space is separable as a topological space . ${\displaystyle L^{p}}$

example

The Borelian algebras ${\displaystyle \sigma }$ im are separable, because they are generated by the cuboids with rational endpoints (or also by the dyadic unit cells ). ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle (a_{1},b_{1})\times \dotsb \times (a_{n},b_{n})}$

literature

• Jürgen Elstrodt: Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin, Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .