Sigma Algebra

Introduction to Sigma Algebras

Recommended Prerequesites

1. Probability

Definition

Let \(\mathcal{X}\) be a set of elements \(\left\{x_1,\dots,x_n\right\}\). Let \(\mathcal{F}\) be a sigma algebra on \(\mathcal{X}\). Then \(\mathcal{F}\) is a collection of subsets of X that satisfies the following properties:

1. Closed under complement
2. Closed under countable unions

We will go further into each of these criteria. For brevity of examples, we will give a value to n. Let \(\mathcal{X}\) be the set \(\left\{x_1, x_2\right\}\). Our sigma algebra for the example shall be $$\mathcal{G}=\{\emptyset,\{x_1\},\{x_2\}, \{x_1, x_2\}\}$$

Closed Under Complement

Since a sigma algebra is closed, then that means for a subset \(A\in \mathcal{F}\), then \(X\setminus A \in \mathcal{F}\).

Example

For our example, we can see that for \(\left\{x_1, x_2\right\}\), $$\{x_1\}\in \mathcal{G} \implies \{x_1, x_2\}\setminus \{x_1\}=\{x_2\}\in \mathcal{G}$$

Closed Under Countable Unions

Let A be a set that is formed from the union of elements of \(\mathcal{F}\), then \(A\in\mathcal{F}\). One distinction is that A need not be formed through a finite number of unions, just countable unions.
Written more formally:
Let \( X \) be a non-empty set, and let \( \mathcal{F} \) be a sigma algebra on \( X \). We say that \( \mathcal{F} \) is closed under countable unions if for any sequence of sets \( A_1, A_2, A_3, \dots \in \mathcal{F} \), the union of these sets is also in \( \mathcal{F} \). In math notation:
$$A_1, A_2, A_3, \dots \in \mathcal{F},$$ then: $$\bigcup_{n=1}^{\infty} A_n \in \mathcal{F}$$

Example

Let \(\mathcal{G}\) be a sigma algebra on the set \(\{x_1, x_2, x_3\}\) If the elements \(\{x_1\},\{x_2\},\{x_3\}\) are in \(\mathcal{G}\), then that implies that all the following:

1. \(\{x_1,x_2\}\in\mathcal{G}\)
2. \(\{x_1,x_3\}\in\mathcal{G}\)
3. \(\{x_2,x_3\}\in\mathcal{G}\)
4. \(\{x_1,x_2,x_3\}\in\mathcal{G}\)

Other Properties

Sometimes you will see closed under countable intersection listed in the definition of a sigma-algebra. This is true, but it can be proven using De Morgan's laws that the two listed properties imply closure under countable intersection.

Examples of Sigma Algebras

Let \(\mathcal{X}\) be the set \(\{x_1, x_2, x_3\}\). It is important to note that there is not just one possible sigma algebra on this set. The following are some possible examples:

1. \(\mathcal{F}_1 = \left\{ \emptyset, \{x_1, x_2, x_3\} \right\}\)
2. \(\mathcal{F}_2 = \left\{ \emptyset, \{x_1\}, \{x_1, x_2, x_3\}, \{x_2, x_3\} \right\}\)
3. \(\mathcal{F}_3 = \left\{ \emptyset, \{x_1\}, \{x_2\}, \{x_3\}, \{x_1, x_2\}, \{x_1, x_3\}, \{x_2, x_3\}, \{x_1, x_2, x_3\} \right\}\)

This last example has a special name: Power Set. As an exercise, come up with another possible sigma algebra on \(\mathcal{X}\).
As another exercise, verify that each of the above examples satisfies the criteria for being a sigma algebra on \(\mathcal{X}\).

Power Set

The Power Set of X is the set of all possible subsets. Written out for our recurring example: $$\mathcal{P}(\{x_1, x_2, x_3\}) = \left\{ \emptyset, \{x_1\}, \{x_2\}, \{x_3\}, \{x_1, x_2\}, \{x_1, x_3\}, \{x_2, x_3\}, \{x_1, x_2, x_3\} \right\}$$ If X is countably infinite, then Cantor's Theorem states that the \(\mathcal{P}\{X\}\) is uncountably infinite.

Borel Sigma Algebra on the Reals

Let X be \(\mathbb{R}\), then the Borel Sigma Algebra on the Reals, \(\mathcal{B}(\mathbb{R})\), is the collection of all sets that can be formed from countable union, countable intersection, and complement of open intervals on \(\mathcal{R}\).
\(\mathcal{B}(\mathbb{R})\) also contains closed intervals and half-open intervals

Sigma Algebra Practice Problems

1. Write out \(\mathcal{P}(\{x_1, x_2\})\).
2. Write out \(\mathcal{P}(\{x_1, x_2, x_3, x_4\})\).
3. \(\text{Given the set } X = \{a, b, c\}, \text{ list all subsets that must be included in a sigma algebra } \mathcal{F} \text{ if } \{a\} \in \mathcal{F}.\)
4. \(\text{Let } X = \{x_1, x_2, x_3\} \text{ and consider the collection of sets:}\)
   \(\mathcal{F} = \left\{ \emptyset, \{x_1\}, \{x_2, x_3\}, \{x_1, x_2, x_3\} \right\}.\)
   \(\text{Prove whether or not } \mathcal{F} \text{ is a sigma algebra on } X.\)
5. \(\text{Let } X = \{1, 2, 3, 4\} \text{ and consider the set } A = \{1, 2\}.\)
   \( \text{Find the smallest sigma algebra } \mathcal{F} \text{ on } X \text{ that contains the set } A.\)
6. \(\text{Let } X = \{x_1, x_2, x_3, x_4\} \text{ and suppose that } \mathcal{F} \text{ is a sigma algebra containing the sets } \{x_1, x_2\} \text{ and } \{x_3\}.\) \(\text{List all the sets that must belong to } \mathcal{F}.\)
7. \(\text{Let } X = \mathbb{N} \text{ (the set of natural numbers), and consider the collection of sets:}\)
   \(\mathcal{F} = \left\{ A \subseteq \mathbb{N} \mid A \text{ is finite, or } A^c \text{ the complement of } A \text{ in } \mathbb{N}) \text{ is finite} \right\}.\)
   \(\text{Prove that } \mathcal{F} \text{ is closed under countable unions, that is, if } A_1, A_2, A_3, \dots \in \mathcal{F},\)
   \(\text{then } \bigcup_{n=1}^{\infty} A_n \in \mathcal{F}.\)