σ ring

A σ-ring or σ-set ring is a special set system that plays an important role in measure theory . A σ-ring is a σ-union stable system of quantities, which is also closed with regard to the formation of differences .

definition

Be any set. A system of sets on , i.e. a set of subsets of , is called a σ-ring (over ) if the following properties are met: ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$

1. ${\displaystyle \emptyset \in {\mathcal {R}}}$: The σ-ring contains the empty set .
2. ${\displaystyle A_{1},A_{2},A_{3},...\in {\mathcal {R}}\Rightarrow \bigcup _{i\in \mathbb {N} }A_{i}\in {\mathcal {R}}}$(Stability / isolation with regard to countable associations ).
3. ${\displaystyle A,B\in {\mathcal {R}}\Rightarrow A\setminus B\in {\mathcal {R}}}$(Stability / seclusion with regard to difference ).

Examples

• A simple example of a σ-ring is that it is the smallest possible σ-ring. ${\displaystyle \{\emptyset \}}$

• Another example is the power set , it is the largest possible σ-ring over a given set . ${\displaystyle {\mathcal {P}}(\Omega )}$ ${\displaystyle \Omega }$

• If there is an arbitrary system of sets over the set , then is ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \Omega }$

${\displaystyle {\mathcal {R}}_{\mathcal {S}}:=\bigcap _{\scriptstyle {\mathcal {S}}\subseteq {\mathcal {R}}' \atop {\scriptstyle {\mathcal {R}}'{\text{ ist σ-Ring}} \atop \scriptstyle {\text{über }}\Omega }}{\mathcal {R}}'}$

the σ-ring generated by . It is the smallest σ-ring over that contains. ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {S}}}$

• The system of all countable subsets of a basic set , i.e. the set system ${\displaystyle \Omega }$

${\displaystyle \{A\subseteq \Omega \mid A{\text{ ist endlich oder abzählbar unendlich }}\}}$,

is a σ ring over . If the basic set is uncountable , this system is not a σ-algebra . ${\displaystyle \Omega }$

properties

In a σ-ring, countable averages are again contained in the σ-ring, because it holds

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

for every sequence in the σ-ring. ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$

This means that finite cuts and unions are also contained in the σ-ring. Likewise, for every set sequence in the σ-ring , Limes superior and Limes inferior of the set sequence are again in : ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}}$

${\displaystyle \liminf _{n\rightarrow \infty }A_{n}\in {\mathcal {R}}\;}$and . ${\displaystyle \;\limsup _{n\rightarrow \infty }A_{n}\in {\mathcal {R}}}$

Furthermore, every countable union of arbitrary sets can be written as a countable union of disjoint sets . This is particularly important for the investigation of set functions for σ-additivity . ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}}$

Operations

Averages of σ-rings

The intersection of two σ-rings and over is always a σ-ring again. Because are , so is ${\displaystyle {\mathcal {R}}_{1}\cap {\mathcal {R}}_{2}}$ ${\displaystyle {\mathcal {R}}_{1}}$ ${\displaystyle {\mathcal {R}}_{2}}$ ${\displaystyle \Omega }$ ${\displaystyle A,B\in {\mathcal {R}}_{1}\cap {\mathcal {R}}_{2}}$

• ${\displaystyle A\setminus B\in {\mathcal {R}}_{1}}$, there , as well ${\displaystyle A,B\in {\mathcal {R}}_{1}}$
• ${\displaystyle A\setminus B\in {\mathcal {R}}_{2}}$, there . ${\displaystyle A,B\in {\mathcal {R}}_{2}}$

Thus , the mean of the σ-rings is also differential stable. The stability with regard to the countable unions follows analogously. ${\displaystyle A\setminus B\in {\mathcal {R}}_{1}\cap {\mathcal {R}}_{2}}$

The statement also applies to the intersection of any number of σ-rings , since the above argument can then be extended to all of these σ-rings. Thus, if there is an arbitrary index set and if all σ-rings are over the same basic set , then the intersection of all these σ-rings is again a σ-ring over : ${\displaystyle \Omega }$ ${\displaystyle I}$ ${\displaystyle {\mathcal {R}}_{i}}$ ${\displaystyle i\in I}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {R}}_{I}}$ ${\displaystyle \Omega }$

${\displaystyle {\mathcal {R}}_{I}:=\bigcap _{i\in I}{\mathcal {R}}_{i}}$.

Associations of σ-rings

The union of two σ-rings and over is generally no longer a σ-ring. For example, consider the two σ-rings ${\displaystyle {\mathcal {R}}_{1}\cup {\mathcal {R}}_{2}}$ ${\displaystyle {\mathcal {R}}_{1}}$ ${\displaystyle {\mathcal {R}}_{2}}$ ${\displaystyle \Omega }$

${\displaystyle {\mathcal {R}}_{1}=\{\emptyset ,\{1\},\{2,3\},\{1,2,3\}\}}$

such as

${\displaystyle {\mathcal {R}}_{2}=\{\emptyset ,\{2\},\{1,3\},\{1,2,3\}\}}$

about so is ${\displaystyle \Omega =\{1,2,3\}}$

${\displaystyle {\mathcal {R}}_{1}\cup {\mathcal {R}}_{2}=\{\emptyset ,\{1\},\{2\},\{1,3\},\{2,3\},\{1,2,3\}\}}$.

However, this set system is not union-stable because it does not contain, and therefore also no σ-ring. ${\displaystyle \{1\}\cup \{2\}=\{1,2\}}$

Products of σ-rings

If and σ-rings are above or , then the product of and is generally no longer a σ-ring (above ). Because if you look at the σ-ring ${\displaystyle {\mathcal {R}}_{1}}$ ${\displaystyle {\mathcal {R}}_{2}}$ ${\displaystyle \Omega _{1}}$ ${\displaystyle \Omega _{2}}$ ${\displaystyle {\mathcal {R}}_{1}\times {\mathcal {R}}_{2}}$ ${\displaystyle {\mathcal {R}}_{1}}$ ${\displaystyle {\mathcal {R}}_{2}}$ ${\displaystyle \Omega _{1}\times \Omega _{2}}$

${\displaystyle {\mathcal {R}}=\{\emptyset ,\{1\},\{2\},\{1,2\}\}}$,

about , the system of sets contains both the sets ${\displaystyle \Omega =\{1,2\}}$ ${\displaystyle {\mathcal {R}}\times {\mathcal {R}}}$

${\displaystyle A=\{1,2\}\times \{1,2\}=\{(1,1),(1,2),(2,1),(2,2)\}}$as well . ${\displaystyle B=\{2\}\times \{2\}=\{(2,2)\}}$

The amount

${\displaystyle A\setminus B=\{(1,1),(1,2),(2,1)\}}$

is not included in, however , because it cannot be represented as the Cartesian product of two sets of . The product is therefore not differential stable and therefore also not a σ-ring. ${\displaystyle {\mathcal {R}}\times {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}}$

Trace of a σ-ring

The trace of a σ-ring with respect to a set , i.e. the set system ${\displaystyle {\mathcal {R}}}$ ${\displaystyle U}$

${\displaystyle {\mathcal {R}}|_{U}:=\{A\cap U\mid A\in {\mathcal {R}}\}}$

is always a σ-ring, regardless of the choice of . ${\displaystyle U}$

Relationship to related structures

Hierarchy of the quantity systems used in measure theory

σ-algebras

A σ-ring that contains the basic set is a σ-algebra (and therefore also an algebra ). Thus every σ-algebra is a σ-ring, but the converse is generally wrong. An example of a σ-ring that is not a σ-algebra is the σ-ring mentioned last in the Examples section above. ${\displaystyle \Omega }$

Rings

Every σ-ring is a ring and thus also a half-ring and a set . The inversions generally do not hold. An example of a ring that is not a σ-ring would be the set system of all finite subsets with a countably infinite basic set.

δ-rings

Every σ-ring is also always a δ-ring , because as shown in the Properties section, σ-rings are always stable with regard to countable sections. Conversely, however, δ-rings are generally not σ-rings. For example, consider any countable set and define the set system of all finite sets on it ${\displaystyle \Omega }$

${\displaystyle {\mathcal {E}}:=\{E\subseteq \Omega \mid |E|<\infty \}}$,

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.

Monotonous classes

Any ring that is a monotonic class is a σ-ring. Because if the quantities are contained in the ring, so is it ${\displaystyle A_{1},A_{2},A_{3},...}$

${\displaystyle B_{n}:=\bigcup _{i=1}^{n}A_{i}}$

due to the properties of the ring contained in the system of quantities. However, the quantities form a monotonically increasing sequence of quantities , hence their limit value ${\displaystyle B_{n}}$

${\displaystyle \lim _{n\to \infty }B_{n}=\bigcup _{n=1}^{\infty }A_{n}}$

due to the properties of the monotonic class also included in the system of sets. The system of sets is thus closed with respect to countable unions. Thus the monotonic class generated by a ring is always a σ-ring.

Conversely, every σ-ring is always a monotonic class due to its stability with regard to countable unions and cuts.

literature

• Jürgen Elstrodt : Measure and integration theory. 4th, corrected edition. Springer, Berlin et al. 2005, ISBN 3-540-21390-2
• Achim Klenke: Probability Theory. 2nd Edition. Springer-Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-76317-8