I went though half a dozen books on measure theory, and it occurs to me that the definition of atom is not particularly unified.
Version $1$: A set $E$ in a $\sigma$-algebra $\Sigma$ is called atom, if there is no proper subset $F\subsetneq E$ such that $F \in \Sigma$.
Alternately:
Version $2$: Let $(X, \Sigma, \mu)$ be a measure space. A set $E ∈ Σ$ is an atom if $\mu(E) > 0$ and whenever $F ∈ Σ, F ⊆ E$ then $\mu(F) = 0$ or $\mu(E \setminus F) = 0$.
The definitions don't seem to be equivalent at least because the first one doesn't even mention measure.
So how should we understand them?
EDIT:
Here is what I've found on MO in the comments to this question:
Pete L. Clark asks:
I know what an atom is in a measure space. What is the definition of an atom in a sigma algebra?
Jonas Meyer responds:
I think it's a nonempty element of the sigma algebra whose only proper subset in the sigma algebra is $∅$.
So the definitions seem to be about different objects. Are they even connected is some way? Does one of them imply another?