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.



When does a probability measure take all values in the unit interval?
A measure space $(\mathbb{P},\Omega,\mathcal{A})$ is atomless if for all $A\in\mathcal{A}$ with $\mathbb{P}(A)>0$ there exists $B\subset A, B\in\mathcal{A}$ such that $0<\mathbb{P}(B)<\mathbb{P}(A)$. Now according to a theorem of Sierpinski, the values of an atomless measure space form an interval. In particular, for probability spaces, every value in $[0,1]$ is taken. The original source of the article can be found [here](http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3125.pdf) (in french). For a proof in english, you can look at on 215D on page 46 in Fremlin's book [Measure Theory 2](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.116.7816&rep=rep1&type=pdf). 


