could someone clarify the following, please?
About the difference between an algebra and a sigma-algebra, is it correct that if $\Omega$ is finite and any possible operation on it is finite, then it is also a sigma-algebra, while if countable operations are possible but are not in the algebra, then it is just an algebra?
Moreover, if $\Omega$ is infinite, like the set of Reals or positive integers, is it still possible to construct an algebra on it? I mean, if $\Omega$ is infinite, then any finite subset in it would have an infinite complement (for example as a countable union of sets); therefore, I would say that if $\Omega$ is infinite, it can't be an algebra but only a sigma-algebra, is that correct?
I hope to have been clear!
Algebra and sigma-algebra
elementary-set-theory
Best Answer
On a finite set $\Omega$ any algebra (of sets) is automatically a $\sigma$-algebra of sets, yes, as infinite operations like countable unions always reduce to finite ones.
On an infinite subset, say $\Omega= \Bbb N$ we can both have algebras that are not $\sigma$-algebras and "proper" $\sigma$-algebras.
In the latter category we have the powerset $\mathscr{P}(\Omega)$, e.g. and in the former we have the algebra of all subsets $A$ of $\Omega$ that are finite or such that $\Omega \setminus A$ is finite. This is clearly not a $\sigma$-algebra, and the examples work for any infinite set $\Omega$.