Algebraic Structure (Measure Theory)

Kinds

Sigma-Algebra

Definition

A sigma-algebra $\mathcal{F}$ is a Family of Sets satisfying the following properties

• $\Omega \in \mathcal{F}\text{or}\mathcal{F}\ne \varnothing$
• $\forall A\in \mathcal{F},A^{\complement }\in \mathcal{F}$
• $\forall \{A_i{\}}_{i=1}^{\infty }\subset \mathcal{F},\bigcup _{i=1}^{\infty }{A}_i\in \mathcal{F}$

where $\Omega$ is a universal set

Intersection of Sigma-Fields

Consider a set of sigma-fields ${\mathcal{F}}_i$ on $\Omega$,

• ${\mathcal{F}}_1\cap {\mathcal{F}}_2$ is a sigma-field on $\Omega$.
• $\bigcup _{i=1}^{\infty }{\mathcal{F}}_i$ is a sigma-field on $\Omega$.
• $\bigcup ^{\infty }{\mathcal{F}}_i$ is a sigma-field on $\Omega$

Facts

If a set of subsets $\mathcal{A}$ of a universal set $\Omega$ is both Pi-System and Lambda-System, then $\mathcal{A}$ is a sigma-algebra on $\Omega$.

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Algebra of Sets

Definition

An algebra $\mathcal{A}$ is a Family of Sets satisfying the following properties

• $\Omega \in \mathcal{A}$
• $\forall A,B\in \mathcal{A},A-B\in \mathcal{A}$
• $\forall A,B\in \mathcal{A},A\cup B\in \mathcal{A}$

where $\Omega$ is a universal set

The conditions imply:

• $\varnothing \in \mathcal{R}$.

Equivalents

• $\Omega \in \mathcal{A}$
• $\forall A\in \mathcal{A},A^{\complement }\in \mathcal{A}$
• $\forall A,B\in \mathcal{F},A\cap B\in \mathcal{A}$

• $\Omega \in \mathcal{A}$
• $\forall A\in \mathcal{A},A^{\complement }\in \mathcal{A}$
• $\forall A,B\in \mathcal{F},A\cup B\in \mathcal{A}$

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Ring of Sets (Measure Theory)

Definition

A ring $\mathcal{R}$ is a Family of Sets satisfying the following properties

• $\forall A,B\in \mathcal{R},A-B\in \mathcal{R}$
• $\forall A,B\in \mathcal{R},A\cup B\in \mathcal{R}$

This implies the following

• $\varnothing \in \mathcal{R}$
• $\forall A,B\in \mathcal{R},A\cap B\in \mathcal{R}\because A\cap B=A-(A-B)$

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Sigma-Ring

Definition

A sigma-ring $\mathcal{R}$ is a Family of Sets satisfying the following properties

• $\forall A,B\in \mathcal{R},A-B\in \mathcal{R}$
• $\forall \{A_i{\}}_{i=1}^{\infty }\subset \mathcal{R},\bigcup _{i=1}^{\infty }{A}_i\in \mathcal{R}$

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Semi-Algebra

Definition

A semi-algebra $\mathcal{A}$ is a Family of Sets satisfying the following properties

• $\Omega \in \mathcal{A}$
• $\forall A,B\in \mathcal{A},A\cap B\in \mathcal{A}$
• $\forall A,B\in \mathcal{A},\exists \{B_i{\}}_{i=1}^n\subset \mathcal{A}\text{s.t.}A-B=\underset{i=1}{\overset{n}{⨄}}{B}_i\text{or}\forall A\in \mathcal{A},\exists \{B_i{\}}_{i=1}^n\subset \mathcal{A}\text{s.t.}{A}^{\complement }=\underset{i=1}{\overset{n}{⨄}}{B}_i$

where $\Omega$ is a universal set

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Semiring

Definition

A semiring $\mathcal{S}$ is a Family of Sets satisfying the following properties

• $\varnothing \in \mathcal{S}$
• $\forall A,B\in \mathcal{S},A\cap B\in \mathcal{S}$
• $\forall A,B\in \mathcal{S},\exists \{B_i{\}}_{i=1}^n\subset \mathcal{S}\text{s.t.}A-B=\underset{i=1}{\overset{n}{⨄}}{B}_i$

where $\uplus$ is a union of the pairwise disjoint sets

Examples

The all below are the semiring on $\Omega =\mathbb{R}$

• $\mathcal{A}=\{(a,b]:-\infty <a<b<\infty \}\cup \{\varnothing \}$
• $\mathcal{A}=\{[a,b):-\infty <a<b<\infty \}\cup \{\varnothing \}$

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Pi-System

A $\pi$-system $\mathcal{P}$ is a Family of Sets satisfying the following property

• $\forall A,B\in \mathcal{P},A\cap B\in \mathcal{P}$

Examples

The all below are the pi-system on $\Omega =\mathbb{R}$

• $\mathcal{A}=\{(a,b]:-\infty <a<b<\infty \}\cup \{\varnothing \}$
• $\mathcal{A}=\{[a,b):-\infty <a<b<\infty \}\cup \{\varnothing \}$
• $\mathcal{A}=\{(a,b):-\infty <a<b<\infty \}\cup \{\varnothing \}$
• $\mathcal{A}=\{[a,b]:-\infty <a<b<\infty \}\cup \{\varnothing \}$

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Transclude of Lambda-System

Summary

	$A\cap B$	$\varnothing$	$A-B$	$\bigcup _i{A}_i\to \underset{i}{⨄}{B}_i$	$\Omega$	$A^c$	$A\cup B$	$\bigcup _{i=1}^{\infty }{A}_i$	$\underset{i=1}{\overset{\infty }{⨄}}{B}_i$	$\bigcap _{i=1}^{\infty }{A}_i$
$\pi$-system	$O$	$X$	$X$	$X$	$X$	$X$	$X$	$X$	$X$	$X$
semi-ring	$O$	$O$	$\Delta$	$O$	$X$	$X$	$X$	$X$	$X$	$X$
semi-algebra	$O$	$O$	$\Delta$	$O$	$O$	$\Delta$	$X$	$X$	$X$	$X$
ring	$O$	$O$	$O$	$O$	$X$	$X$	$O$	$X$	$X$	$X$
algebra	$O$	$O$	$O$	$O$	$O$	$O$	$O$	$X$	$X$	$X$
$\sigma$-ring	$O$	$O$	$O$	$O$	$X$	$X$	$O$	$O$	$O$	$O$
$\lambda$-system	$X$	$O$	$\Delta$’	$X$	$O$	$O$	$X$	$X$	$O$	$X$
$\sigma$-field	$O$	$O$	$O$	$O$	$O$	$O$	$O$	$O$	$O$	$O$

${\Delta }^{\prime }$: $A\subset B\Rightarrow B-A\in \mathcal{D}$

Diagrams

flowchart TD
 
subgraph ALGEBRA
sia["$\sigma$-algebra"]
a[algebra]
sea[semialgebra]
 
sia --> a
a --> sea
end
 
subgraph RING
sir["$\sigma$-ring"]
r[ring]
ser[semiring]
 
sir --> r
r --> ser
end
 
subgraph LAMBDA
ls["$\lambda$-system"]
end
 
ps["$\pi$-system"]
 
sia --> sir
sia --> ls
 
a --> r
 
sea --> ser
 
ser --> ps

flowchart TD
 
ps["$\pi$-system"]
 
subgraph RING
ser[semiring]
r[ring]
sir["$\sigma$-ring"]
 
ser -->|"$\cup$-stable"|r
r -->|"$\sigma$-$\cup$-stable"|sir
end
 
subgraph ALGEBRA
sea[semialgebra]
a[algebra]
sia["$\sigma$-algebra"]
 
sea -->|"$\cup$-stable"|a
a -->|"$\sigma$-$\cup$-stable"|sia
end
 
subgraph LAMBDA
ls["$\lambda$-system"]
 
ls -->|"$\cap$-stable"|sia
end
 
ps -->|"$\uplus$-stable"|ser
 
ser -->|"$\Omega$-contained"|sea
r -->|"$\Omega$-contained"|a
sir -->|"$\Omega$-contained"|sia