Glossary

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Sigma algebra

A sigma algebra is family of subsets, ℱ say, of some set Ω such that
1. Ω∈ℱ,
2. If A∈ℱ then Ac=Ω∖A∈ℱ,
3. If Ak, k=1,2,...∈ℱ then ⋃    kAk∈ℱ.
So a sigma algebra is a family of subsets closed under taking complements and taking countable unions. This will also lead to that a sigma algebra is closed under countable intersections by using De Morgan's laws. This construction is used to describe events which we can assign probababilties to in a consistant way.
For the set of real numbers the most common sigma algebra is the Borel sigma algebra, denoted ℬ(ℝ). This is the smallest sigma algebra that contains all open intervals.