Existence of non-atomic probability measure for given measure zero sets Let $\Omega$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $\Omega$. Let $N$ be a collection of measurable subsets of $\Sigma$. 
Question: What conditions on $\Sigma$ and $N$ guarantee that there exists a non-atomic probability measure  $\mu:\Sigma\to [0,1]$ such that for any $E\in \Sigma$ if $\mu(E)=0$,  then $E\in N$ ?  
Edited to make question coherent.
 A: Thanks Michael Greinecker and commenter. 
The main practical problem for me in applying commenter's idea was that the weak $\sigma$-distributive property in Maharam's 1947 paper, in  Kelley's paper, and in   Todorcevic's amazing paper of 2004 on measure algebras may not hold if we choose an arbitrary $\sigma$-ideal $J$ in $N$ (and it certainly has no clear meaning for what I am doing). 
In the end, the best fit for my work was Ryll-Nardzewski's result published in the addendum section of Kelley and not Kelley's result with the distributive property. 
1) There exists a sequence $B_n$ of families of subsets of $\Sigma$ such that
$(\Sigma\setminus N)\subseteq \bigcup_{n} B_n$.
2) Each $B_n$ has a positive intersection number (as in Kelley).
3) Each $B_n$ is open for increasing sequences; (if $E_m\uparrow E\in B_n$, then eventually $E_m\in B_n$). 
The final condition (3) of Nardzewski guarantees that $\Sigma\setminus \bigcup_{n} B_n$ is a $\sigma$-ideal. 
Condition (2) guarantees that there is  a finitely additive (positive) probability measure $\nu_n$ on $\Sigma$ that is bounded away from zero on $B_n$.
Condition (3) tells us that from $\nu_n$ we can define a countably additive probability measure $\mu_n$ that also measures elements of $B_n$ positively. 
Letting $\mu= \sum_{n=1}^\infty 2^{-n} \mu_n$, we have the required measure.      
For the converse suppose that $\mu$ is the required measure, letting 
$B_n=\{ \mu>1/n\}$ we see that (1), (2), and (3); hold.   
