Countable additivity
Topic of Probability spaces and events, part of Stochastic processes
Countable additivity
| Type | Topic |
| Taught in | Probability spaces and events |
| Part | Stochastic processes |
The axiom that probability adds over countably many disjoint events: \mathrm{P}(\bigcup_n A_n)=\sum_n\mathrm{P}(A_n). Finite additivity is not enough for limits of events; countable additivity gives continuity along monotone sequences and the Borel–Cantelli lemmas.
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).
Used in
See also
Event · Borel–Cantelli lemmas · Probability space · Generated and Borel sigma-algebras · Limits of sequences of events · Sigma-algebra