Does every non empty open set has measure greater than zero?
Yes, take a point $x \in A$, with $A$ open. Then exists $\epsilon > 0$ such that $(x-\epsilon, x+\epsilon) \subset A$. So: $${\frak m}(x-\epsilon,x+\epsilon) \leq {\frak m}A \implies 0 < 2\epsilon \leq {\frak m}A \implies {\frak m}A > 0.$$
Every non-empty open set contains at least one open interval, and open intervals have a positive measure.