Probability space
Definition taught in Probability spaces and events, part of Stochastic processes
Probability space
| Type | Definition |
| Taught in | Probability spaces and events |
| Part | Stochastic processes |
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).
The triple (\Omega,\mathcal{F},\mathrm{P}) is a probability space. The probability of an event A is written \mathbb{P}(A) as well as \mathrm{P}(A).
Used in
- Taught in Probability spaces and events, where the derivation and examples are.
- All 3 questions on Probability spaces and events