$0$ on finite sets, $\infty$ on infinite ones: finitely but not countably additive

Let $\Omega$ be a countably infinite set and let F consist of all subsets of $\Omega$. Define $\mu(A)=0$ if A is finite, $\infty$ if A is infinite. Which of the following statements are true? (i) $\mu$ is finitely additive; (ii) $\mu$ is countably additive; (iii) $\{A_n\}$ is an $\uparrow$ sequence of sets with $\mu(A_n)=0$ $\forall n$, but $\mu(\Omega)=\infty$.