Given a measure space $(\Omega, \mathcal{F}, \mu)$, we say that $\mu$ is non-atomic if for any measurable set $A$ with positive (> $0$) measure there exists a measurable subset, $B$, of A such that $\mu(A) > \mu (B) > 0$.
On the other hand, we say that $\mu$ is $s$-finite (which is weaker than $\sigma$-finite) if we can write it as $\mu(A) =\sum_{n=1}^{\infty}\xi_{n}(A)$ for every measurable set $A$, with $\xi_{n}$ finite measures on $(\Omega, \mathcal{F})$ for every $n$.
Studying these strucutres, I tried but couldn't find examples of: 1) a non-atomic measure which is not a s-finite measure, and a measure which is not non-atomic nor s-finite.
It would be extremely helpful if anyone has some examples of both classes (and references for classifications of s-finite measures too, if possible).