Sigma-algebra/Definition
From Maths
Note: This page is intended to allow transclusion of the definition (so one copy is used many times), for the definition of [ilmath]\sigma[/ilmath]-algebra see sigma-algebra
Definition
Given a set [ilmath]X[/ilmath] a [ilmath]\sigma[/ilmath]-algebra on [ilmath]X[/ilmath] is a family of subsets of [ilmath]X[/ilmath], [ilmath]\mathcal{A} [/ilmath][Note 1], such that[1]:
- [ilmath]\forall A\in\mathcal{A}[A^C\in\mathcal{A}][/ilmath] - Stable under complements
- [ilmath]\forall\{A_n\}_{n=1}^\infty\subseteq\mathcal{A}\left[\bigcup_{n=1}^\infty A_n\in\mathcal{A}\right][/ilmath] - Stable under countable union
Notes
- ↑ So [ilmath]\mathcal{A}\subseteq\mathcal{P}(X)[/ilmath]
