If A and B are disjoint subsets of real numbers, and one of them is measurable can we say
m\*(A U B)=m\*(A)+m\*(B)?
I am unable to find counter example. I feel this is not true.



about measure theory
This is true. If for example A is measurable it is measurable in the sense of Caratheodory so that  

For every set C we will have 
$m\*(C) = m\*(C\cap A) + m\*(C \setminus A)$.


This with $C=A \cup B$ is your assertion


