Sigma-Algebra

Compact study note.

Summary

One sigma-algebra is the collection of events on which a probability measure is allowed to operate. It is closed under complements and countable unions, so probability rules remain stable under repeated event operations.^[1]

Prerequisites

• Sample Space and Events

Notation and Assumptions

Collection $\mathcal{F}\subseteq \mathcal{P}(\mathrm{\Omega })$ is sigma-algebra when it contains $\mathrm{\Omega }$ , is closed under complements, and is closed under countable unions:

Essential Result

One correct finite example is $\mathcal{F}=\mathcal{P}(\{1,2,3\})$ . On $[0,1]$ , the collection of all intervals is not one sigma-algebra because countable unions of intervals need not be intervals.

Small Example

With $\mathrm{\Omega }=\{H,T\}$ , $\mathcal{F}=\{\mathrm{\varnothing },\{H\},\{T\},\mathrm{\Omega }\}$ is a sigma-algebra. Complements and countable unions stay inside the same collection.

Common Mistakes

• Saying 'all intervals' form one sigma-algebra.
• Forgetting countable closure; finite closure alone defines only an algebra of sets.

Connections

• Borel Sigma-Algebra
• Probability Space
• Random Variable

References