I was wondering, can the following theorem be true for finitely additive measures defined on algebras not $\sigma$-algebras. (Theorem is in Bogachev's Measure Theory Vol I).
I was not sure about it, but I found a coubterexample for nonatomic measure, which is: $\lambda=\mu+2\nu$, where $\mu$ is the Lebesgue measure on the unit intrval $I$, and $\nu$ is a 0−1-valued finitely additive measure on $I$ such that $\nu(A)=0$ if $\mu(A)=0$. $\lambda$ is a nonatomic finitely additive measure, but has not partition for $0<\varepsilon<2$.
I really need a result like this: suppose $\mu$ is a finitely additive nonatomic measure on a countable Boolean algebra $B$. Then, for every $\varepsilon>0$, there is a finite partition $P$ of the unit element such that $\mu(p_i)<\varepsilon$" for each $p_i\in P$ and $i=1,\dots,n$.