Let $\mu$ be a positive finite measure in some $\sigma$-field $\cal{S}$ of subsets of $X$. How find a decomposition $X=B \cup C$ on the disjoint sum of two measurable sets $B, C$ such that the measure $\mu|_B$ is atomless (i.e. it does not contain atoms) and $\mu|_{C}$ is purely atomic (i.e every measurable subset of $C$ of positive measure is the sum of finite on countable many atoms), where, for fixed $Z \in \cal{S}$, $\mu|_Z (E)=\mu(E \cap Z)$ for every $E \in \cal{S}$.
I think that if there exist a finite number of disjoint atoms, we can find decomposition in the following way.
Let $B_1$ be an arbitrary atom (if there exists),and assume that we have defined atoms $B_1,...,B_n$ such that $B_n \subset X\setminus \bigcup_{i=1}^{n-1}B_i$. We denote by $B_{n+1}$ an atom contained in $X\setminus \bigcup_{i=1}^{n}B_i$ if there exist. In the case when there is a finite number atoms this sequence is finite. We take $B=\bigcup B_n$, $C=X\setminus B$ and we have desired decomposition.
I don't sure, how to find the decomposition in the case when there exist a countable number of disjoints atoms? Does it suffice to take $B=\bigcup_{n=1}^\infty B_n$, $C=X\setminus B$?