Meaning of countable in definitions of sigma algebra

[Rasalhague]

Meaning of "countable" in definitions of sigma algebra

In the third axiom defining a $\sigma$-algebra, $(X,\Sigma)$, does countable mean (a) "finite or countably infinite", or does it mean (b) "countably infinite".

 

[mathman]

In this context it means (a), i.e. every countable(finite or countably infinite) union or intersection of sets in Σ is in Σ.

 

[Rasalhague]

Thanks.

 

[g_edgar]

In this context it means (a), i.e. every countable(finite or countably infinite) union or intersection of sets in Σ is in Σ.

I agree. But in this case (because the empty set is already there) it doesn't matter. A finite union is the same as a countably infinite union by appending empty sets to the list.

 

[blue_raver22]

In the context of sigma algebra, the term "countable" typically refers to the second meaning (b) - that is, countably infinite. This means that the collection of sets in the sigma algebra is either finite or countably infinite, and does not include uncountably infinite sets. This is important in the definition of sigma algebra as it ensures that the collection of sets is well-defined and manageable for mathematical operations. So, in short, countable in the definition of sigma algebra means that the collection of sets is countably infinite.