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+---
+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 %}
+
+## Generated {{ page.title }}
+
+{% 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 $X$.
+
+## Products of {{ page.title }}
+