Does there exist a measure $\mu$ on $([0,1],\mathfrak B_{[0,1]})$ such that $\mu(\text {Cantor Set})=1$ and $\mu (\{x\})=0$ for every $x \in [0,1]$

Question

Does there exist a measure $\mu$ on $([0,1],\mathfrak B_{[0,1]})$ such that $\mu(\text {Cantor set})=1$ and $\mu (\{x\})=0$ for every $x \in [0,1]$?

If there is no assumption on measure of singleton we can take $\mu$ to be Counting measure but it does not with the assumptions.I know that the Lebesgue measure of Cantor set is zero,i think Riesz Representation theorem should help here but i am unable to see it.Any ideas?