δ-system (measure theory)

A δ-system , also called a σ- -stable system ${\displaystyle \cap }$ , is a set system in measure theory that is closed with regard to certain set-theoretic operations. The addition "δ-" is also used in more specific set systems to indicate that it is also a δ system, for example δ-rings or δ-algebras.

definition

A set system on the base set is called a δ-system if it is closed with respect to countable averages. So it applies to that ${\displaystyle {\mathcal {S}}\subset {\mathcal {P}}(\Omega )}$ ${\displaystyle \Omega }$ ${\displaystyle A_{1},A_{2},A_{3},...\in {\mathcal {S}}}$

${\displaystyle \bigcap _{i=1}^{\infty }A_{i}\in {\mathcal {S}}}$

is.

Examples

• Every averagely stable set system with finitely many elements is a δ-system. To do this, in the sequence of the sets of the set system, all sets whose index exceeds a certain value are simply set as a fixed set. As a result, the cut is formally a countable sequence, but it only contains a finite number of different sets.
• Every σ-algebra is by definition a δ-system.

properties

• Every δ-system that is stable with respect to complement formation is also automatically stable with respect to countable unions. This follows from De Morgan's laws .
• The reverse is also true: every system that is stable towards countable unions and complement formation is a δ-system.

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 .