Measure Theory Note

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
 

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Measure

Definition

A function defined on a measurable space $(\Omega ,\mathcal{F})$, in which satisfies the following conditions

• Non-negativity: $\mu :\mathcal{F}\to {\mathbf{R}}_0^{+}$
• $\mu (\varnothing )=0$
• Countable additivity: $\mu (\underset{i}{⨄}{A}_i)=\sum \mu (A_i)$

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

Kinds

• Probability Measure
• Finite Measure
• Sigma-Finite Measure
• Counting measure
• Dirac measure

probability measure $\subset$ finite measure $\subset$ $\sigma$-finite measure

Summary

	$\mu (\varnothing )=0$	$\sigma$-add	$A_i\uparrow \Omega$, $\mu (A_i)<\infty$	$\mu (\Omega )<\infty$	$\mu (\Omega )=1$		monotone	$\sigma$-subadd	conti-below	conti-above
msr	$O$	$O$	$X$	$X$	$X$		$O$	$O$	$O$	$\Delta$
$\sigma$-finite-msr	$O$	$O$	$O$	$X$	$X$		$O$	$O$	$O$	$\Delta$
finite-msr	$O$	$O$	$O$	$O$	$X$		$O$	$O$	$O$	$O$
prob-msr	$O$	$O$	$O$	$O$	$O$		$O$	$O$	$O$	$O$

Notations

• $\sigma$-additive: $\mu (\underset{i=1}{\overset{\infty }{⨄}}{B}_i)={\sum }_{i=1}^{\infty }\mu (B_i)$
• $\sigma$-subadditive: $\mu (\bigcup _{i=1}^{\infty }{A}_i)\le \sum _{i=1}^{\infty }\mu (A_i)$
• monotone: $A\subset B\Rightarrow \mu (A)\le \mu (B)$
• continuous from below: $A_i\uparrow A\Rightarrow \mu (\underset{n\to \infty }{\lim }{A}_i)=\underset{n\to \infty }{\lim }\mu (A_i)$
• continuous from above: $[A_i\downarrow A\text{and}\mu (A_1)<\infty ]\Rightarrow \mu (\underset{n\to \infty }{\lim }{A}_i)=\underset{n\to \infty }{\lim }\mu (A_i)$

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Measurable Space

Definition

Consider a set $X$ and the sigma-filed defined on the set $X$ $(X,\mathcal{A})$

In contrast to a Measure Space, no Measure is needed for a measurable space

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Measure Space

Definition

A measure space consists of a Measurable Space together Measure on it

$(X,\mathcal{A},\mu )$ where $X$ is a set, $\mathcal{A}$ is a Sigma-Field defined on the set $X$, and $\mu$ is a Measure.

Kinds

• Probability Space
• Sigma-Finite Measure Space

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Sigma-Field Generated by A Families of Sets

Definition

$\forall \mathcal{A}\subset 2^{\Omega },\exists !,\sigma (\mathcal{A})$ Intersections of sigma-fields are a sigma-field, and the Intersection of all Sigma-Field containing $\mathcal{A}$ is the smallest Sigma-Field containing $\mathcal{A}$. Thus, there exist the unique smallest Sigma-Field containing every set in the given Family of Sets $\mathcal{A}$.

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Dynkin's Pi-Lambda Theorem

Definition

$l(\mathcal{P})=\sigma (\mathcal{P})$ where $\mathcal{P}$ is a Pi-System

If $\mathcal{P}$ is a Pi-System, then the small Lambda-System containing $\mathcal{P}$, $l(\mathcal{P})$ is the sigma-field generated by $\mathcal{P}$, $\sigma (\mathcal{P})$.

$\sigma (\mathcal{P})\subset \mathcal{L}$ where $\mathcal{P}$ is a Pi-System and $\mathcal{L}$ is a Lambda-System containing $\mathcal{P}$

If $\mathcal{P}$ is a Pi-System and $\mathcal{L}$ is the Lambda-System containing $\mathcal{P}$, then $\sigma (\mathcal{P})\subset \mathcal{L}$

Proof

If $\mathcal{A}$ is a both Lambda-System and Pi-System, then $\mathcal{A}$ is a Sigma-Field

Since the condition of $\sigma (\mathcal{P})$ is more complex, then $l(\mathcal{P})$, $l(\mathcal{P})\subset \sigma (\mathcal{P})$ is satisfied

The $l(\mathcal{P})$ containing a Pi-System $\mathcal{P}$ is a Pi-System itself. If Lambda-System and Pi-System, then Sigma-Field Therefore Lambda-System $l(\mathcal{P})$ is a Sigma-Field

Since $\sigma (\mathcal{P})$ is the smallest Sigma-Field, $l(\mathcal{P})\supset \sigma (\mathcal{P})$ is satisfied

Summary

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

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Extension from Pi-System

Definition

Probability Measure

If $\mathcal{A}$ is a Pi-System on the Probability Space $(\Omega ,\sigma (\mathcal{A}),P)$, then the Probability Measure $P:\sigma (\mathcal{A})\to [0,1]$ is uniquely determined by the probability measure-like function on the Pi-System $P^{\ast }:\mathcal{A}\to [0,1]$

Finite Measure

If $\mathcal{A}$ is a Pi-System containing the universal set $\Omega$ on the finite Measure Space $(\Omega ,\sigma (\mathcal{A}),\mu )$, then the Finite Measure $\mu :\sigma (\mathcal{A})\to [0,\mu (\Omega )<\infty ]$ is uniquely determined by the finite measure-like function on the Pi-System ${\mu }^{\ast }:\mathcal{A}\to [0,\mu (\Omega )<\infty ]$

Sigma-Finite Measure

If $\mathcal{A}$ is a Pi-System on the Sigma-Finite Measure Space $(\Omega ,\sigma (\mathcal{A}),\mu )$, then the Sigma-Finite Measure $\mu :\sigma (\mathcal{A})\to [0,\infty ]$ is uniquely determined by the sigma-finite measure-like function on the Pi-System ${\mu }^{\ast }:\mathcal{A}\to [0,\infty ]$

Facts

If the two measures ${\mu }_1,{\mu }_2$ defined on a Sigma-Field Generated by A Families of Sets $\sigma (\mathcal{A})$ satisfy the following conditions, then they are the same ${\mu }_1={\mu }_2$.

• $\forall A\in \mathcal{A}:{\mu }_1(A)={\mu }_2(A)$
• ${\mu }_1(\Omega )={\mu }_2(\Omega )<\infty$

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Caratheodory Extension Theorem

Definition

If $\mathcal{A}$ is a Semiring on $\Omega$, and the function (pre-measure) $\tilde{m}:\mathcal{A}\to [0,\infty ]$ satisfies the following conditions, then the function $\tilde{m}$ uniquely extended to the Measure on $(\Omega ,\sigma (\mathcal{A}))$.

Conditions

• $\tilde{m}(\varnothing )=0$
• additivity: $\tilde{m}(\underset{i=1}{\overset{n}{⨄}}{B}_i)={\sum }_{i=1}^n\tilde{m}(B_i)$
• $\sigma$-sub additivity: $\tilde{m}(\bigcup _{i=1}^{\infty }{A}_i)\le {\sum }_{i=1}^{\infty }\tilde{m}(A_i)$
• $\sigma$-finite: $\exists \{A_i{\}}_{i=1}^{\infty }\subset \mathcal{A},\text{s.t.}m(A_i)<\infty ,\bigcup _{i=1}^{\infty }{A}_i=\Omega$

The conditions 1~3: $\tilde{m}$ satisfies measure-like conditions on $\mathcal{A}$ The condition 4: $\tilde{m}$ satisfies Sigma-Finite Measure-like condition on $\mathcal{A}$, it makes the extension result unique.

The set $\mathcal{A}$ should be Semiring. So, the conditions of this theorem are stronger than Extension from Pi-System, but can be applied for every Measure.

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Usual Topology

Definition

$\mathcal{U}=\left\{O|O={\bigcup }_{i=1}^{\infty }(a_i,b_i)\text{s.t.}(a_i,b_i\in \Omega \text{and}{a}_i\le b_i)\right\}$

$\mathcal{U}$ is a Family of Sets whose elements can be expressed as a countable union of open intervals.

Facts

The usual topological space $(\mathbb{R},\mathcal{U})$ satisfies $\forall o\in O,\exists a,b\in \mathbb{R},o\in (a,b)\subset O$ where $O$ is the set of open intervals

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Borel Sigma-Field

Definition

$\mathcal{R}=\mathcal{B}(X):=\sigma (\mathcal{T})$

Let $(X,\mathcal{T})$ be a Topological Space, where $X$ is the set and $\mathcal{T}$ is the topology on $X$. The Borel sigma-field $\mathcal{B}(X)$ is the smallest Sigma-Field that contains all open sets in $\mathcal{T}$.

Borel Sigma-Field on the Real Number

$\mathcal{R}=\mathcal{B}(\mathbb{R}):=\sigma (\mathcal{U})$ where $\mathcal{U}$ is Usual Topology on $\mathbb{R}$.

Borel Sigma-Field on the Extended Real Number Space

$\bar{\mathcal{R}}=\mathcal{B}(\bar{\mathbb{R}})$ where $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\infty ,\infty \}$.

Facts

Every element of $\mathcal{B}(\mathbb{R})$ is measurable by Lebesgue Measure

• ${\mathcal{A}}_1:=\{A\subset \mathbb{R}:A\text{ is U-open}\}=\mathcal{U}$
• ${\mathcal{A}}_2:=\{(a,b):a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_3:=\{[a,b):a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_4:=\{(a,b]:a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_5:=\{[a,b]:a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_6:=\{(-\infty ,b):a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_7:=\{(-\infty ,b]:a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_8:=\{(a,\infty ):a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_9:=\{[a,\infty ):a,b\in \mathbb{R},a<b\}$

$\mathcal{R}:=\mathcal{B}(\mathbb{R})=\sigma ({\mathcal{A}}_1)=\sigma ({\mathcal{A}}_2)=\cdots =\sigma ({\mathcal{A}}_9)$ The above sets are all the Borel sigma-field.

Proof Every element of $\sigma (\mathcal{A})$ is generated by the elements of $\mathcal{A}$ and the complementary set and countable union by the conditions of Sigma-Field If $\sigma ({\mathcal{A}}_1)$ can be generated by ${\mathcal{A}}_1$ and the operations, and ${\mathcal{A}}_1$ can be generated by ${\mathcal{A}}_2$ and the operations, then ${\mathcal{A}}_2$ also generate $\sigma ({\mathcal{A}}_1)$ Therefore, $\sigma ({\mathcal{A}}_1)=\sigma ({\mathcal{A}}_2)$

An element of Borel sigma-field $B\in \mathcal{B}(X)$ is called Borel set.

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Lebesgue Measure

Definition

A Measure on Borel Sigma-Field, which is a unique extension of function $\tilde{m}:\mathcal{A}\to [0,\infty ]$ on a Semiring $\mathcal{A}=\{(a,b]:a,b\in \mathbb{R},a<b\}$ by Caratheodory Extension Theorem

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