Sigma-algebra
Topic of Probability spaces and events, part of Stochastic processes
Sigma-algebra
| Type | Topic |
| Taught in | Probability spaces and events |
| Part | Stochastic processes |
A collection \mathcal{F} of subsets of \Omega that contains \Omega and is closed under complements and countable unions, and hence under countable intersections. The events of a probability space are the members of a sigma-algebra, so that the countable operations used for limits of events stay inside the collection.
In Probability spaces and events
A sigma-algebra \mathcal{F} on \Omega contains \Omega, is closed under complements, and is closed under countable unions. It is then closed under countable intersections, by taking complements. A probability measure \mathrm{P}:\mathcal{F}\to[0,1] satisfies \mathrm{P}(\Omega)=1 and is countably additive: for pairwise disjoint events A_n\in\mathcal{F},
\mathrm{P}\Bigl(\bigcup_{n=1}^{\infty} A_n\Bigr)=\sum_{n=1}^{\infty}\mathrm{P}(A_n).
In Probability spaces and events, continued
Probability has to say which outcomes and which events are being discussed. On a finite set every subset can be an event. On an uncountable set, a sigma-algebra is a collection large enough for the countable operations the later arguments use.
Used in
See also
Probability space · Probability measure · Event · Borel–Cantelli lemmas · Generated and Borel sigma-algebras · Limits of sequences of events · Countable additivity