A basic question on probability measure Can we have a probability measure in $[0,1]$ such that it assigns $0$ to every singleton but for other sets it assigns either $0$ or $1$. Hint Enough.  

You seem to be looking for a two-valued probability measure that is defined for all subsets of $[0,1]$. The set $[0,1]$ has the same cardinality as the set of infinite sequences of $0$'s and $1$'s. It is more convenient to work with that set. 
Let $A_n$ be the set of sequences that are $0$ at $n$, and let $B_n$ be the set of sequences that are $1$ at $n$. Then one of $A_n$ or $B_n$ has measure $1$. Call it $C_n$. The intersection of the $C_n$ has measure $1$. But this intersection is a singleton. 
