---
title: σ-Algebras
parent: Measure Theory
grand_parent: Measure and Integration
nav_order: 1
---

# {{ page.title }}

{% definition Sigma-Algebra, Measurable Space, Measurable Set %}
A *σ-algebra* on a set $X$ is a collection $\mathcal{A}$ of subsets of $X$ such that

- $X$ belongs to $\mathcal{A}$,
- if $A \in \mathcal{A}$, then $X \setminus A \in \mathcal{A}$,
- the union of any countable subcollection of $\mathcal{A}$ belongs to $\mathcal{A}$.

A *measurable space* is a pair $(X,\mathcal{A})$ consisting of
a set $X$ and a σ-algebra $\mathcal{A}$ on $X$. \
The subsets of $X$ belonging to $\mathcal{A}$ are called *measurable sets*.
{% enddefinition %}

{% example %}
On every set $X$ we have the σ-algebras $\braces{\varnothing,X}$ and $\mathcal{P}(X)$.
{% endexample %}

{% proposition %}
If $\mathcal{A}$ is *σ-algebra* on a set $X$, then:

- $\varnothing$ belongs to $\mathcal{A}$,
- if $A,B \in \mathcal{A}$, then $B \setminus A \in \mathcal{A}$,
- the intersection of any countable subcollection of $\mathcal{A}$ belongs to $\mathcal{A}$.
{% endproposition %}

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{% proposition Intersection of σ-Algebras %}
If $\braces{\mathcal{A}_i}$ is a family of σ-algebras on a set $X$,
then $\bigcap_i \mathcal{A}_i$ is a σ-algebra on $X$.
{% endproposition %}

{% definition Generated σ-Algebras %}
Suppose $\mathcal{E}$ is any collection of subsets of a set $X$.
The *σ-algebra generated by $\mathcal{E}$*, denoted by $\sigma(\mathcal{E})$, is
defined to be the intersection of all σ-algebras on $X$ containing $\mathcal{A}$.
{% enddefinition %}

By the previous proposition, $\sigma(\mathcal{E})$ is in fact a σ-algebra on $X$.

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