I look for a example of family of atomic measures such that their sum is not atomic. A measure $\mu$ on a $\sigma$-algebra $S$ of subsets of $X$ is called atomic if every measurable set of positive measure contain an atom. Here by an atom we mean a set set $E\in S$ with $\mu(E)>0$ such that for every $F\subset E$ with $F\in S$ we have $\mu(F)=0$ or $\mu(E\setminus F)=0$).
I found the following example (page 651, Example 1.3) but I did not understand everything.
Assume that $X$ is a compact Hausdorff space in which all singletons are not in $G_\delta$ (Examples of such topologies are here). Let $S$ be a $\sigma$ -algebra generated by all compact $G_\delta$ subsets of $X$. For each $x \in X$ we define a measure $\mu_x$ on $S$ by putting $\mu_x(E)=1$ if $x \in E$ and $\mu_x(E)=0$ otherwise. Let $\mu$ be a measure on $S$ defined by $\mu(\emptyset)=0$ and $\mu(E)=\infty$ otherwise.
How to show that $\mu(E)=\sum \{\mu_x(E): x \in X\}$ for all $E\in S$?
It is clear that it suffices to know that every nonempty $E\in S$ is infinite.
Thanks for reading and I would appreciate it if you could solve my problem.