Atomic intersection of sigma-fields and some of its consequences

Let $(Omega,mathcal{F},P)$ be a probability space. For each $mathcal{G}subsetmathcal{F}$, define $overline{mathcal{G}}$ as the $sigma$-field generated by $mathcal{G}$ and those sets $Finmathcal{F}$ satisfying $P(F)in{0,1}$. Conditions for $P$ to be atomic on $cap_{i=1}^koverline{mathcal{A}_i}$, with $mathcal{A}_1,dots,mathcal{A}_ksubsetmathcal{F}$ sub-$sigma$-fields, are given. Conditions for $P$ to be 0-1-valued on $cap_{i=1}^koverline{mathcal{A}_i}$ are given as well. These conditions are useful in various fields, including Gibbs sampling, iterated conditional expectations and the intersection property.