From 98236a8eb89c09174971fcb28360cf1ea2b9a8e4 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 23 Oct 2021 19:16:13 +0200
Subject: Expand knowledge base

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+---
+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.
+
+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.
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