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

Question

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

score 1 · Answer 1 · answered Aug 18 '18 at 18:45

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By Stone's representation theorem for Boolean algebras, the theory of algebras on sets coincides essentially with the theory of Boolean algebras in general. As such, general algebras of sets are extremely useful.

answered Aug 18 '18 at 18:45

Michael Greinecker

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

1 Answers1