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To start with measure theory, we want to handle the abstract setting of a measure space at first. This definitions should lead to a formal understanding of abstract measure theoretical background. The most important notion is that of a <math>\sigma</math>-Algebra. | |||
{{definitioncard|<math>\sigma</math>-Algebra and measurable sets| | |||
Let <math>E</math> be a Set. A <math>\sigma</math>-Algebra <math>\mathcal{A}</math> on <math>E</math> is a collection of subsets of <math>E</math>, which satisfies the following conditions. | |||
<ul style{{=}}"list-style-type:lower-roman"><li>The ground space has to be in <math>\A</math>, i.e. <math>E\in\mathcal{A}</math>, | |||
</li> | |||
<li>If <math>A\in\mathcal{A}</math> then <math>A^C\in\mathcal{A}</math>, where <math>A^C</math> denotes the complement of <math>A</math>, | |||
</li> | |||
<li>If <math>(A_n)_{n\in\mathbb{N}}\subset \mathcal{A}</math> is a collection of elements in <math>\A</math> then <math>\bigcup_{n\in\mathbb{N}}A_n\in\mathcal{A}</math>. | |||
</li> | |||
</ul> | |||
Moreover, the elements of <math>\mathcal{A}</math> are called measurable sets. The tupel <math>(E,\A)</math>, that is the set <math>E</math> endowed with the <math>\sigma</math>-Algebra <math>\A</math>, is called a measurable space. }} | |||
{{alert-info | This definition implies the following. | |||
<ul style{{=}}"list-style-type:lower-roman"><li>Every <math>\sigma</math>-Algebra <math>\A</math> is a subset of <math>\mathcal{P}(E)</math>, i.e. <math>\mathcal{A}\subseteq\mathcal{P}(E)</math>, where <math>\mathcal{P}(E)</math> denotes the power set of <math>E</math>, that is the set of all subsets of <math>E</math>. | |||
</li> | |||
<li>The empty set has to be in <math>\A</math>, i.e. <math>\varnothing\in\mathcal{A}</math>, | |||
</li> | |||
<li>If <math>(A_n)_{n\in\N}\subset\mathcal{A}</math> is a collection of elements of <math>\A</math> then <math>\bigcap_{n\in\mathbb{N}}A_n\in\mathcal{A}</math>, i.e. | |||
<math display="block"> | |||
\bigcap_{n\in\mathbb{N}}A_n=\left(\bigcup_{n\in\mathbb{N}}A_n^C\right)^C. | |||
</math> | |||
</li> | |||
</ul> | |||
}} | |||
'''Example''' | |||
[Examples of <math>\sigma</math>-Algebras] | |||
We give the following simple examples for <math>\sigma</math>-Algebras on a set <math>E</math>. | |||
<ul style{{=}}"list-style-type:lower-roman"><li><math>\mathcal{A}=\{\emptyset,E\}</math> is called the ''trivial'' or the ''smallest'' <math>\sigma</math>-Algebra on <math>E</math>. | |||
</li> | |||
<li><math>\mathcal{A}=\mathcal{P}(E)</math> is the ''largest'' <math>\sigma</math>-Algebra{{efn|This is convenient for finite and countable measureable spaces}} on <math>E</math>. | |||
</li> | |||
<li><math>\mathcal{A}=\{A\subset E\mid A</math> is countable or <math>A^C</math> is countable<math>\}</math>. | |||
</li> | |||
</ul> | |||
\begin{exer} | |||
Show that the examples above are indeed <math>\sigma</math>-Algebras. | |||
\end{exer} | |||
Let us consider a set <math>A_n\in \A</math> for <math>n\in\mathbb{N}</math>. The following observation are useful | |||
<ul style{{=}}"list-style-type:lower-roman"><li>If <math>A_n</math> is a countable set for all <math>n\in\N</math>, then <math>\bigcup_{n\in\mathbb{N}}A_n</math> is also a countable set and we know that | |||
<math display="block"> | |||
\bigcup_{n\in\mathbb{N}}A_n\in\mathcal{A}. | |||
</math> | |||
</li> | |||
<li>If there is a <math>n_0\in\N</math> such that <math>A_{n_0}</math> is an uncountable set, it follows that <math>A_{n_0}^C</math> is a countable set, i.e. | |||
<math display="block"> | |||
\left(\bigcup_{n\in\mathbb{N}}A_n\right)^C=\bigcap_{n\in\mathbb{N}}A_n^C\subset A_{n_0}^C, | |||
</math> | |||
which implies that <math>\left(\bigcup_{n\in\N}A_n\right)^C</math> is countable. | |||
</li> | |||
</ul> | |||
We can construct many more interesting <math>\sigma</math>-Algebras by noting that any arbitrary intersection of <math>\sigma</math>-Algebras is again a <math>\sigma</math>-Algebra. Let therefore <math>(\mathcal{A}_i)_{i\in I}</math> be a family of <math>\sigma</math>-Algebras and <math>I</math> an arbitrary Indexset, then the set | |||
<math display="block"> | |||
\mathcal{A}:=\bigcap_{i\in I}\mathcal{A}_i | |||
</math> | |||
is also a <math>\sigma</math>-Algebra. | |||
{{definitioncard|Generated <math>\sigma</math>-Alegbra| | |||
Let <math>E</math> be a set and let <math>\mathcal{C}</math> be a subset of <math>\mathcal{P}(E)</math>. Then there exists a smallest <math>\sigma</math>-Algebra, denoted by <math>\sigma(\mathcal{C})</math>, which contains <math>\mathcal{C}</math>. This <math>\sigma</math>-Algebra may be defined as | |||
<math display="block"> | |||
\sigma(\mathcal{C})=\bigcap_{\mathcal{C}\subset\mathcal{A}\atop\mathcal{A} \text{a $\sigma$-Algebra} }\mathcal{A}. | |||
</math> | |||
}} | |||
{{alert-info | We can observe that if <math>\mathcal{C}</math> is a <math>\sigma</math>-Algebra itself, then clearly <math>\sigma(\mathcal{C})=\mathcal{C}</math>. Moreover, for two subsets <math>\mathcal{C}\subset\mathcal{P}(E)</math> and <math>\mathcal{C}'\subset\mathcal{P}(E)</math> with <math>\mathcal{C}\subset\mathcal{C}'</math> we get that <math>\sigma(\mathcal{C})\subset\sigma(\mathcal{C}')</math>. | |||
}} | |||
'''Example''' | |||
Let <math>E</math> be a set and let <math>A\subset E</math> be a subset. Moreover, let <math>\mathcal{C}=A</math>. Then we would get | |||
<math display="block"> | |||
\sigma(\mathcal{C})=\{\varnothing,A,A^C,E\}. | |||
</math> | |||
More generally, let <math>E=\bigcup_{i\in I}E_i</math>, where <math>I</math> is a finite or countable index set and <math>E_i\cap E_j=\varnothing</math> for <math>i\not=j</math>. Then we call <math>(E_i)_{i\in I}</math> a partition of <math>E</math> and the set | |||
<math display="block"> | |||
\mathcal{A}=\left\{\bigcup_{j\in J}E_j\mid J\subset I\right\} | |||
</math> | |||
has the structure of a <math>\sigma</math>-Algebra. Now let <math>\mathcal{C}=\left\{\{x\}\mid x\in E\right\}</math>. Then we would get that | |||
<math display="block"> | |||
\sigma(\mathcal{C})=\left\{A\subset E\mid \text{$A$ is countable or $A^C$ is countable}\right\}. | |||
</math> | |||
==General references== | |||
{{cite arXiv|last=Moshayedi|first=Nima|year=2020|title=Lectures on Probability Theory|eprint=2010.16280|class=math.PR}} | |||
==Notes== | |||
{{notelist}} |
Latest revision as of 01:53, 8 May 2024
To start with measure theory, we want to handle the abstract setting of a measure space at first. This definitions should lead to a formal understanding of abstract measure theoretical background. The most important notion is that of a [math]\sigma[/math]-Algebra.
Let [math]E[/math] be a Set. A [math]\sigma[/math]-Algebra [math]\mathcal{A}[/math] on [math]E[/math] is a collection of subsets of [math]E[/math], which satisfies the following conditions.
- The ground space has to be in [math]\A[/math], i.e. [math]E\in\mathcal{A}[/math],
- If [math]A\in\mathcal{A}[/math] then [math]A^C\in\mathcal{A}[/math], where [math]A^C[/math] denotes the complement of [math]A[/math],
- If [math](A_n)_{n\in\mathbb{N}}\subset \mathcal{A}[/math] is a collection of elements in [math]\A[/math] then [math]\bigcup_{n\in\mathbb{N}}A_n\in\mathcal{A}[/math].
Moreover, the elements of [math]\mathcal{A}[/math] are called measurable sets. The tupel [math](E,\A)[/math], that is the set [math]E[/math] endowed with the [math]\sigma[/math]-Algebra [math]\A[/math], is called a measurable space.
- Every [math]\sigma[/math]-Algebra [math]\A[/math] is a subset of [math]\mathcal{P}(E)[/math], i.e. [math]\mathcal{A}\subseteq\mathcal{P}(E)[/math], where [math]\mathcal{P}(E)[/math] denotes the power set of [math]E[/math], that is the set of all subsets of [math]E[/math].
- The empty set has to be in [math]\A[/math], i.e. [math]\varnothing\in\mathcal{A}[/math],
- If [math](A_n)_{n\in\N}\subset\mathcal{A}[/math] is a collection of elements of [math]\A[/math] then [math]\bigcap_{n\in\mathbb{N}}A_n\in\mathcal{A}[/math], i.e.
[[math]] \bigcap_{n\in\mathbb{N}}A_n=\left(\bigcup_{n\in\mathbb{N}}A_n^C\right)^C. [[/math]]
Example
[Examples of [math]\sigma[/math]-Algebras] We give the following simple examples for [math]\sigma[/math]-Algebras on a set [math]E[/math].
- [math]\mathcal{A}=\{\emptyset,E\}[/math] is called the trivial or the smallest [math]\sigma[/math]-Algebra on [math]E[/math].
- [math]\mathcal{A}=\mathcal{P}(E)[/math] is the largest [math]\sigma[/math]-Algebra[a] on [math]E[/math].
- [math]\mathcal{A}=\{A\subset E\mid A[/math] is countable or [math]A^C[/math] is countable[math]\}[/math].
\begin{exer} Show that the examples above are indeed [math]\sigma[/math]-Algebras. \end{exer} Let us consider a set [math]A_n\in \A[/math] for [math]n\in\mathbb{N}[/math]. The following observation are useful
- If [math]A_n[/math] is a countable set for all [math]n\in\N[/math], then [math]\bigcup_{n\in\mathbb{N}}A_n[/math] is also a countable set and we know that
[[math]] \bigcup_{n\in\mathbb{N}}A_n\in\mathcal{A}. [[/math]]
- If there is a [math]n_0\in\N[/math] such that [math]A_{n_0}[/math] is an uncountable set, it follows that [math]A_{n_0}^C[/math] is a countable set, i.e.
[[math]] \left(\bigcup_{n\in\mathbb{N}}A_n\right)^C=\bigcap_{n\in\mathbb{N}}A_n^C\subset A_{n_0}^C, [[/math]]which implies that [math]\left(\bigcup_{n\in\N}A_n\right)^C[/math] is countable.
We can construct many more interesting [math]\sigma[/math]-Algebras by noting that any arbitrary intersection of [math]\sigma[/math]-Algebras is again a [math]\sigma[/math]-Algebra. Let therefore [math](\mathcal{A}_i)_{i\in I}[/math] be a family of [math]\sigma[/math]-Algebras and [math]I[/math] an arbitrary Indexset, then the set
is also a [math]\sigma[/math]-Algebra.
Let [math]E[/math] be a set and let [math]\mathcal{C}[/math] be a subset of [math]\mathcal{P}(E)[/math]. Then there exists a smallest [math]\sigma[/math]-Algebra, denoted by [math]\sigma(\mathcal{C})[/math], which contains [math]\mathcal{C}[/math]. This [math]\sigma[/math]-Algebra may be defined as
Example
Let [math]E[/math] be a set and let [math]A\subset E[/math] be a subset. Moreover, let [math]\mathcal{C}=A[/math]. Then we would get
More generally, let [math]E=\bigcup_{i\in I}E_i[/math], where [math]I[/math] is a finite or countable index set and [math]E_i\cap E_j=\varnothing[/math] for [math]i\not=j[/math]. Then we call [math](E_i)_{i\in I}[/math] a partition of [math]E[/math] and the set
has the structure of a [math]\sigma[/math]-Algebra. Now let [math]\mathcal{C}=\left\{\{x\}\mid x\in E\right\}[/math]. Then we would get that
General references
Moshayedi, Nima (2020). "Lectures on Probability Theory". arXiv:2010.16280 [math.PR].