open sets and measurability

Let $A$ be a subset of $mathbb{R}$ with $m^*(A)<infty$ (where $m^*$ denotes the Lebesgue outer measure). Is it true that for every open subset $Osubset mathbb{R}$ we always have the equality $m^*(O)=m^*(Ocap A)+m^*(Ocap A^c)$ ? Does anybody have an idea or clue? Thanks in advance.