1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
|
---
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 applications
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**.
<hr>
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 $\omega \in \Omega$.
<hr>
The concept of information can be extended for
stochastic processes (i.e. time-indexed random variables):
if $\{ X_t : t \ge 0 \}$ is a stochastic process,
its **filtration** $\mathcal{F}_t$ contains all
the information generated by $X_t$ up to the current time $t$:
$$\begin{aligned}
\mathcal{F}_t
= \sigma(X_s : 0 \le s \le t)
\end{aligned}$$
In other words, $\mathcal{F}_t$ is the "accumulated" $\sigma$-algebra
of all information extractable from $X_t$,
and hence grows with time: $\mathcal{F}_s \subset \mathcal{F}_t$ for $s < t$.
Given $\mathcal{F}_t$, all values $X_s$ for $s \le t$ can be computed,
i.e. if you know $\mathcal{F}_t$, then the present and past of $X_t$ can be reconstructed.
Given some filtration $\mathcal{H}_t$, a stochastic process $X_t$
is said to be *"$\mathcal{H}_t$-adapted"*
if $X_t$'s own filtration $\sigma(X_s : 0 \le s \le t) \subseteq \mathcal{H}_t$,
meaning $\mathcal{H}_t$ contains enough information
to determine the current and past values of $X_t$.
Clearly, $X_t$ is always adapted to its own filtration.
## References
1. U.F. Thygesen,
*Lecture notes on diffusions and stochastic differential equations*,
2021, Polyteknisk Kompendie.
|
