---
title: "σ-algebra"
firstLetter: "S"
publishDate: 2021-10-22
categories:
- Mathematics

date: 2021-10-18T10:01:35+02:00
draft: false
markup: pandoc
---

# $\sigma$-algebra

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 $\Omega \backslash A \in \mathcal{F}$ 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.


## Notable examples

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**.

Another example of a $\sigma$-algebra is the **information**
obtained by observing a [random variable](/know/concept/random-variable/) $X$.
Let $\sigma(X)$ be the information generated by observing $X$,
i.e. the events whose occurrence can be deduced from the value of $X$:

$$\begin{aligned}
    \sigma(X)
    = X^{-1}(\mathcal{B}(\mathbb{R}^n))
    = \{ A \in \mathcal{F} : A = X^{-1}(B) \mathrm{\:for\:some\:} B \in \mathcal{B}(\mathbb{R}^n) \}
\end{aligned}$$

In other words, if the realized value of $X$ is
found to be in a certain Borel set $B \in \mathcal{B}(\mathbb{R}^n)$,
then the preimage $X^{-1}(B)$ (i.e. the event yielding this $B$)
is known to have occurred.

Given a $\sigma$-algebra $\mathcal{H}$,
a random variable $Y$ is said to be *"$\mathcal{H}$-measurable"*
if $\sigma(Y) \subseteq \mathcal{H}$,
meaning that $\mathcal{H}$ contains at least
all information extractable from $Y$.

Note that $\mathcal{H}$ can be generated by another random variable $X$,
i.e. $\mathcal{H} = \sigma(X)$.
In that case, the **Doob-Dynkin lemma** states
that $Y$ is only $\sigma(X)$-measurable
if $Y$ can always  be computed from $X$,
i.e. there exists a function $f$ such that
$Y(\omega) = f(X(\omega))$ for all $\omega \in \Omega$.



## References
1.  U.F. Thygesen,
    *Lecture notes on diffusions and stochastic differential equations*,
    2021, Polyteknisk Kompendie.