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+---
+title: "Sigma-algebra"
+date: 2021-10-22
+categories:
+- Mathematics
+- Measure theory
+layout: "concept"
+---
+
+In set theory, given a set $\Omega$, a $\sigma$**-algebra**
+is a family $\mathcal{F}$ of subsets of $\Omega$
+with these properties:
+
+1.  The full set is included $\Omega \in \mathcal{F}$.
+2.  For all subsets $A$, if $A \in \mathcal{F}$,
+    then its complement $\Omega \!-\! A \in \mathcal{F}$ too.
+3.  If two events $A, B \in \mathcal{F}$,
+    then their union $A \cup B \in \mathcal{F}$ too.
+
+This forms a Boolean algebra:
+property (1) represents TRUE,
+(2) is NOT, and (3) is AND,
+and that is all we need to define all logic.
+For example, FALSE and OR follow from the above points:
+
+4.  The empty set is included $\varnothing \in \mathcal{F}$.
+5.  If two events $A, B \in \mathcal{F}$,
+    then their intersection $A \cap B \in \mathcal{F}$ too.
+
+For a given $\Omega$, there are typically multiple valid $\mathcal{F}$,
+in which case you need to specify your choice.
+Usually this would be the smallest $\mathcal{F}$
+(i.e. smallest family of subsets)
+that contains all subsets of special interest
+for the topic at hand.
+Likewise, a **sub-$\sigma$-algebra**
+is a sub-family of a certain $\mathcal{F}$,
+which is a valid $\sigma$-algebra in its own right.
+
+A notable $\sigma$-algebra is the **Borel algebra** $\mathcal{B}(\Omega)$,
+which is defined when $\Omega$ is a metric space,
+such as the real numbers $\mathbb{R}$.
+Using that as an example, the Borel algebra $\mathcal{B}(\mathbb{R})$
+is defined as the family of all open intervals of the real line,
+and all the subsets of $\mathbb{R}$ obtained by countable sequences
+of unions and intersections of those intervals.
+The elements of $\mathcal{B}$ are **Borel sets**.
+
+
+
+## References
+1.  U.H. Thygesen,
+    *Lecture notes on diffusions and stochastic differential equations*,
+    2021, Polyteknisk Kompendie.
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