Example of atoms with respect to $ \mu$
Solution 1:
If $A^c$ is finite, then $A$ must be infinite. Otherwise, $T=A\cup A^c$ would be finite. So $\mu(A)=1$. Now, if $E\subseteq A$ and $E\in\mathcal{A}$, then, by the latter condition, $E$ is finite and satisfies $\mu(E)=0$, or $E^c$ is finite, which , again. implies that $E$ is infinite and $\mu(E)$. So either $\mu(E)=\mu(A)$ or $\mu(E)=0$ for all $E\subseteq A$ with $E\in\mathcal{A}$, which is exactly the definition of $A$ being an atom.