I want to prove that $m_{∗}(A \cap (a, b)) = (b − a) − m^* (A^c \cap (a, b))$.
Since $(a,b)$ is open, it is measurable. Then $(b-a)=m^*((a,b))=m_*((a,b)$.
Also by the definition of measurable set $m^*((a,b))=m^*(A \cap (a,b))+m^*(A \cap (a,b)^c)$.
But the original problem includes the complement of set $A$ instead of $(a,b)$. What am I doing wrong in this proof?
It seems like if set $A$ is measurable, then we can easily derive this equality.
Thanks