Measurable Spaces

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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.

Definition ([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.

• 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.

This definition implies the following.

• 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 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

[[math]] \mathcal{A}:=\bigcap_{i\in I}\mathcal{A}_i [[/math]]

is also a [math]\sigma[/math]-Algebra.

Definition (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]] \sigma(\mathcal{C})=\bigcap_{\mathcal{C}\subset\mathcal{A}\atop\mathcal{A} \text{a $\sigma$-Algebra} }\mathcal{A}. [[/math]]

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]] \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]] \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]] \sigma(\mathcal{C})=\left\{A\subset E\mid \text{$A$ is countable or $A^C$ is countable}\right\}. [[/math]]

General references

Moshayedi, Nima (2020). "Lectures on Probability Theory". arXiv:2010.16280 [math.PR].