Monday, 5 November 2007

measure theory - Are there sigma-algebras of cardinality kappa>2aleph0kappa>2aleph0 with countable cofinality?

A standard homework in measure theory textbooks asks the student to prove that there are not countably infinite sigmasigma-algebras. The only proof that I know is via a contradiction argument which yields no estimate on the minimum cardinality of an infinite sigmasigma-algebra.



Given an a set XX of infinite cardinality kappakappa, the sigmasigma-algebra of all co-countable subsets of XX is of cardinality 2kappa2kappa kappaaleph0kappaaleph0. This example doesn't tell me whether there are sigmasigma-algebras of cardinality below 2aleph02aleph0, if I don't assume the Continuum Hypothesis.



My question is as the title says: Are there sigmasigma-algebras of every uncountable cardinality?



Edit: The combined answer with Stephen, Matthew proves that the cardinality of a sigmasigma-algebra is necessarily at least 2aleph02aleph0. Further, for each cardinality kappage2aleph0kappage2aleph0 with uncountable cofinality, the sigmasigma-algebra of countable (or cocountable) subsets of a set XX with cardinality kappakappa, is of cardinality kappakappa.



What is left is whether for kappage2aleph0kappage2aleph0 with cf(kappa)=aleph0cf(kappa)=aleph0 are there sigmasigma-algebras of cardinality kappakappa. (I changed the title to reflect this.)



Thanks Stephen, Matthew, Apollo, for the combined work!

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