Let $(X, \mathcal F, P)$ be a probability space.

# Question

What kind of condition is this: there exists a sequence $(a_n)_n \subseteq X$ such that

$\forall$ measurable $A \subseteq X$, $P(A) > 0 \implies a_n \in A$ for some $n$.

**Context.** I'm reviewing a paper for a conference and the authors assume this condition on the probability space in order to prove a result. I was wondering if this condition has a technical name, etc. in the greater math literature.

Proof of Fedor's claim.We show that the set compliment $X\setminus (a_n)_n$ has zero measure. Indeed, $X \setminus (a_n)_n$ (assumed measurable!) contains no $a_n$, thus by the contrapositive of condition, we must have $P(X\setminus (a_n)_n) = 0$. Thus the support of $P$ is a contained in the sequence $(a_n)_n$, and so the former must be a countable combination of atoms $\Box$. I should probably close the question... $\endgroup$ – dohmatob Aug 9 at 9:41