Finite additivity alone gives superadditivity over countable disjoint unions

Let $\mathbf F$ be a $\sigma$-field of subsets of $\Omega$. Let $\mu$ be a non-negative, finitely additive set function on $\mathbf F$. Suppose $\{A_n\}$ is a sequence of disjoint sets such that $\bigcup_{n=1}^\infty A_n\in\mathbf F$. Then which of the following statements is always correct?