Please be noted that charges are finitely additive measures and measure are countably additive ones.
Theorem 2.1. Let $\mu$ be a charge on a Boolean algebra $B$. Each of the following conditions implies the succeeding condition.
$\quad(\rm i)$ $\mu$ is strongly nonatomic on $B$.
$\;\;\,(\rm ii)$ $\mu$ is strongly continuous on $B$.
$\;\,(\rm iii)$ $\mu$ is nonatomic on $B$.If $\mu$ is a measure and $B$ is a Boolean $\sigma$-algebra then the conditions are equivalent.
for charges $ (i)\Rightarrow (ii) \Rightarrow (iii)$, and they are equivalent for measures. The author said, the implications $ (i)\Rightarrow (ii) \Rightarrow (iii)$ follow from their definitions.
But I am stack in proving $(ii) \Rightarrow (iii)$ ??!! I do not know how to conclude it.
Let $\mu(b) > 0$, and let $b_1,\,\dotsc,\,b_n$ as in definition, where $\varepsilon = \mu(b)/2$.
Consider
$$(b\wedge b_i),\quad 1\leqslant i \leqslant n.$$