σ-Algebra

Topics: Probability

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(definition)

A $\sigma$-algebra (sigma-algebra), denoted $\mathcal{F}$, is a family of subsets of one same sample space $\Omega$ such that:

1. $\Omega \in \mathcal{F}$
2. If $A\in \mathcal{F}⟹A^c\in \mathcal{F}$
3. If $(A_n{)}_{n\ge 1}$ is a collection of subsets in $\mathcal{F}$, then ${\bigcup }_{n=1}^{\infty }{A}_n\in \mathcal{F}$

Example

Let’s toss a coin.

$$\Omega =\{X,C\}$$

Is $\mathcal{F}=\mathcal{P}(\Omega )$ (power set) a $\sigma$-algebra?

Let’s see what we have:

1. $\Omega \in \mathcal{F}$
2. We check each subset and its complement:

• $\varnothing \in \mathcal{F}\wedge {\varnothing }^c=\Omega \in \mathcal{F}$
• $\Omega \in \mathcal{F}\wedge {\Omega }^c=\varnothing \in \mathcal{F}$
• $X\in \mathcal{F}\wedge X^c=C\in \mathcal{F}$
• $C\in \mathcal{F}\wedge C^c=X\in \mathcal{F}$

1. We check every possible union:

• $\varnothing \cup \Omega =\Omega \in \mathcal{F}$
• $\Omega \cup C=\Omega \in \mathcal{F}$
• $C\cup X\cup \varnothing =\Omega \in \mathcal{F}$
• $X\cup \Omega \cup \varnothing \cup C=\Omega \in \mathcal{F}$
• etc.

Since the three axioms are satisfied, $\mathcal{F}=\mathcal{P}(\Omega )$ is a $\sigma$-algebra.