Define the function $\mu: S \rightarrow [0,\infty)$ by $\mu(A) = 0 $ if $A$ is finite and $\mu(A) = \infty$ if $A$ is countable. Show that $\mu $ is finitely additive.
Here's what I have so far. Consider a disjoint sequence $A_1,A_2,...,A_n$ such that $A=\cup_{k =1 }^{n} A_k \in S. $
So the cases I want to examine is that $A$ is finite and $A$ is countable. If $A$ is a finite set, it is similar to adding the probability of two mutually exclusive events.
I am not sure about the second case. Am I on the right track so far?
Thank you for your help!
Only two cases have to be considered.
Case 1): One of the sets is countably infinite. In this case the union is countably infinite. So $\mu (\cup A_i)=\sum \mu(A_i)=\infty$.
Case 2): Each $A_i$ is finite. In this case the union is finite so $\mu (\cup A_i)= \sum \mu(A_i)=0$.