# Sigma-ring

A Sigma-ring or $\sigma$-ring, is closely related to rings of sets, and just like algebras of sets are related to rings of sets, so is $\sigma$-ring to $\sigma$-algebra

## Definition

A non-empty class of sets [ilmath]S[/ilmath] is a [ilmath]\sigma[/ilmath]-ring if[1]:

• for [ilmath]A\in S[/ilmath] and [ilmath]B\in S[/ilmath] then [ilmath]A-B\in S[/ilmath]
• If $\{A_n\}_{n=1}^\infty\subset S$ then $\cup^\infty_{n=1}A_n\in S$

That is to say it is closed under subtraction and countable union

## References

1. Measure Theory, p24 - Halmos - Graduate Texts in Mathematics (18) - Springer