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.  

No: $S_0 := [0, 1]$ has measure $1$. One of $[0, \frac{1}{2}]$ or $[\frac{1}{2}, 1]$ has measure $1$ and the other has measure $0$. Let $S_1$ be the one with measure $1$. Inductively define $S_i$ to be the half of $S_{i - 1}$ with measure $1$. Show $\bigcap_i S_i$ is a singleton with measure $1$.
