Question on non-atomic measures Let $(X,\Sigma,\mu)$ be a finite measure space.  Call $\mu$ nonatomic if for any $A\in\Sigma$ with $\mu(A)>0$ there exists $B\in\Sigma$ with $B\subset A$ and $0<\mu(B)<\mu(A)$. 

Suppose that $\mathcal S$ be the algebra that generates  $\Sigma$. I am asking to what is the relation between nonatomicity of $\mu$ and $\mu|_\mathcal S$.


I'm no measure theory wizard, so check my argument well for errors!
Consider the collection of sets $\mathcal A\subset\Sigma$ defined by: $A\in\mathcal A$ if and only if the following two conditions both hold


*

*$\mu(A)=0$ or $\exists B\in\Sigma:0<\mu(B)<\mu(A)$.

*$\mu(A^c)=0$ or $\exists B\in\Sigma:0<\mu(B)<\mu(A^c)$.


If we assume that $\mu|_\mathcal S$ is 'nonatomic', then $\mathcal A$ is a Dynkin class containing $\mathcal S$ (check this! I found the second definition found on Wikipedia to be simplest to work with). Since $\mathcal S$ is clearly stable under finite intersections, it follows from Dynkin's Lemma that $\Sigma\subset \mathcal A$.
