We know that a σ-algebra is either finite or not countable. Can I find an algebra (field of sets) that is countably infinite? I know that the proof for said theorem uses the fact that a σ-algebra is closed so countable unions, but I cant find an example.

Question

We know that a   σ-algebra is either finite or not countable. Can I find an algebra (field of sets) that is countably infinite? I know that the proof for said theorem uses the fact that a   σ-algebra is closed so countable unions, but I cant find an example.

Jaylee Dodson · Accepted Answer

Let   X  =      N   and define   Σ  =  {  A  ⊆      N    :      |    A      |    &amp;lt;  ∞  }  ∪  {  A  ⊆      N    :      |        N    ∖  A      |    &amp;lt;  ∞  }. You can easily check that this is an algebra. It is not a sigma-algebra because       ⋃          n      =      1        ∞    {  2  n  }} is a countable union of sets in   Σ which doesn&#039;t belong to   Σ.There are countably many finite subsets of       N  , and so also countably many co-finite subsets. Thus |      |    Σ      |    =      ℵ    0  .

Misael Matthews · Accepted Answer

The algebra of finite-or-cofinite subsets of a set of cardinality   κ  ≥      ℵ    0   has cardinality   κ, therefore you can find algebras of any infinite cardinality.

We know that a σ -algebra is either finite or not countable. Can I find an alg

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