Sigma-Algebra

A sigma-algebra (denoted $\mathcal{F}$) on a Sample Space $S$ is a collection of subsets of $S$ that satisfy:

1. $S\in \mathcal{F}$
2. If $A\in \mathcal{F}$, then its complement $A^c$ is also in $\mathcal{F}$.
3. If $A_1,A_2,A_3,\cdots \in \mathcal{F}$, then ${\bigcup }_{i=1}^{\infty }{A}_i\in \mathcal{F}$.

Because of (3), sigma-algebras are automatically closed under:

• countable intersections
• finite unions

Intuition: $\mathcal{F}$ lists exactly the events to which we are allowed to assign probabilities.

The sigma-algebra is the second component of the Probability Space.

Example

Let $S=\{1,2,3\}$. One valid sigma-algebra on $S$ is:

$$\mathcal{F}=\{\varnothing ,\{1,2\},\{3\},S\}$$

Check:

• $S$ is included
• Closed under complement:
  • ${\varnothing }^c=S$
  • $\{1,2{\}}^c=\{3\}$
• Closed under countable unions (only finite unions here):
  • $\varnothing \cup \{1,2\}=\{1,2\}$
  • $\{1,2\}\cup \{3\}=S$

Therefore this collection is a sigma-algebra on $S$.