Such measure cannot exists on any reasonable topological space (e.g. Euclidean space or any space with no isolated points) for the following reason:
By definition a measure has to be (finitely) additive: Measure of the disjoint union of two sets in the sigma algebra must be the sum of their measures.
Now given an open set, can we decompose it into two disjoint sets both with empty interior? The answer is yes for any reasonable spaces, right? (For example, we write R as rationals and irrationals.) If such decomposition exists, we have both sets with measure $0$ but their union has positive measure: $\Rightarrow\Leftarrow $ can't happen~