TITLE: When does a probability measure take all values in the unit interval?
QUESTION [9 upvotes]: Let $\mathbb{P}$ be a probability measure on some probability space $(\Omega,\mathcal{A})$. Are there conditions on the $\sigma$-algebra $\mathcal{A}$ such that for every real number $c\in [0,1]$ we find a set $A\in\mathcal{A}$ with $\mathbb{P}(A)=c$. 
It is like the intermediate value theorem for continuous functions.

REPLY [4 votes]: A necessary and sufficient condition is that every atom is no larger than the sum of all smaller atoms, plus the non-atomic part.