Does the type of an additivity (finite, infinite) imply anything about the elements being added (finite, infinite, countable, uncountable)

Question

I am working with a problem involving an infinite sample space, a sigma-algebra, a measure assigning the value 0 to finite events, and, assigning infinity to infinite events.

I am asked to show that the measure is finitely additive but not countably additive.

I understand what finitely and countably additive means in regards to taking unions, namely:
Finitely additive = "measure is closed under finite unions"
Countably additive = "measure is closed under countable unions"

But, what I don't understand is if it implies anything about the events being taken unions of. In other words, do finitely additive mean "only taking unions of finite events", and, countably additive mean "only taking unions of countable events"?

Thanks

Answer 1

Let $X$ be an infinite space and chose a countable subset of $X$ say $A=\{x_1,x_2,...x_n,...\}$

Then $+\infty=\mu(A)>\sum_{n=1}^{\infty}\mu(\{x_n\})=0$

So the measure is not countably additive.