# Is there anything else than a $\sigma$-algebra or is $\sigma$-algebra the only meaningful algebra on sets?

Is there anything else than a $\sigma$-algebra or is $\sigma$-algebra the only meaningful algebra on sets?

It seems that the $\sigma$-algebra has been invented in order to serve some particular higher level constructs, but I wonder if there are some other useful algebras on sets?

• Regarding your first comment yes. $\sigma$-algebras are very useful in measure theory. Namely, measures are defined over $\sigma$-algebras. – Dog_69 Aug 18 '18 at 18:39
• (Non countable) algebras are useful constructs in many senses. Generally they form the stage for anything involving (finite) unions and complementation. – copper.hat Aug 18 '18 at 18:44
• Boolean Algebras by Roman Sikorski discusses algebras in which "countable" is replaced by various cardinal numbers (not necessarily the same for intersections and unions). See also Is there a $\sigma$-algebra on $\mathbb{R}$ strictly between the Borel and Lebesgue algebras? and papers such as this. – Dave L. Renfro Aug 18 '18 at 20:37