Category:Axioms/Sigma-Algebra Axioms

From ProofWiki
Jump to navigation Jump to search

This category contains axioms related to Sigma-Algebra Axioms.

Let $X$ be a set.

Let $\Sigma$ be a system of subsets of $X$.

$\Sigma$ is a $\sigma$-algebra over $X$ if and only if $\Sigma$ satisfies the sigma-algebra axioms:

\((\text {SA 1})\)   $:$   Unit:    \(\ds X \in \Sigma \)      
\((\text {SA 2})\)   $:$   Closure under Complement:      \(\ds \forall A \in \Sigma:\) \(\ds \relcomp X A \in \Sigma \)      
\((\text {SA 3})\)   $:$   Closure under Countable Unions:      \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) \(\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \)      

Pages in category "Axioms/Sigma-Algebra Axioms"

The following 3 pages are in this category, out of 3 total.