δ-ring (set system)

A δ-ring ( delta-ring ) is a special set system that is used in measure theory , a branch of mathematics that deals with generalized volume concepts .

definition

A δ-ring is a set ring that is also a δ-system , that is, it is a non-empty set system that is stable or closed with regard to union , subtraction and countable sections .

If one writes out all the requirements for a δ-ring individually, they read:

Are included in the δ-ring, so is also included. (Stability in relation to union) ${\ displaystyle A, B}$ ${\ displaystyle A \ cup B}$
Are included in the δ-ring, so is also included. (Stability with regard to difference formation) ${\ displaystyle A, B}$ ${\ displaystyle A \ setminus B}$
Are included in the δ-ring, so is also included. (Stability with regard to countable cuts) ${\ displaystyle A_ {1}, A_ {2}, A_ {3}, \ dots}$ ${\ displaystyle \ bigcap _ {i = 1} ^ {\ infty} A_ {i}}$

Properties and examples

Every σ-ring is also always a δ-ring, because it is true

{\ displaystyle \ bigcap _ {i = 1} ^ {\ infty} A_ {i} = A_ {1} \ setminus \ bigcup _ {i = 2} ^ {\ infty} (A_ {1} \ setminus A_ {i })}

for every sequence of sets contained in the σ-ring .

{\ displaystyle (A_ {i}) _ {i \ in \ mathbb {N}}}

However, the reverse is generally not true. For example, consider any countable set and define the set system of all finite sets on it ${\ displaystyle X}$

{\ displaystyle {\ mathcal {E}}: = \ {E \ subset X \, | \, | E | <\ infty \}}

,

so it is a δ-ring, since countable cuts of finite sets are again finite. But it is not a σ-ring, because countable unions of finite sets are in general not finite.

use

For example, δ-rings are used in modifications of the measure expansion theorem by Carathéodory , where instead of expanding to the σ-algebra generated by a ring, only the generated δ-ring is expanded.

Web links

Allan Cortzen: Delta Ring . In: MathWorld (English).

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 .