Definition of the outer measure Let $X$ be a set. By definition, for every sequence of sets (disjoint or not), an outer measure $\theta:\mathcal{P}X\rightarrow [0,+\infty]$ is a monotic, countably subadditive (hence subadditive) function which vanishes at $0$. 
We then have four possibilities:


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*$A\cap B=\emptyset$ implies $\theta(A\cup B)=\theta A+\theta B$ (additivity)

*$A\cap B=\emptyset$ implies $\theta(A\cup B)<\theta A+\theta B$ (e.g. Bernstein set)

*$A\cap B\not=\emptyset$ implies $\theta(A\cup B)<\theta A+\theta B$ (quite intuitive)
My question is how to interpret the last possibility:


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*$A\cap B\not=\emptyset$ implies $\theta(A\cup B)=\theta A+\theta B$


Is it as well a case of nonmeasurability? 
If not, could anybody provide me with a simple example?

It is not a case of non measurability: take the lebesgue outer measure and take $A = [0,1]$ and $B = [1,2]$.
$A \cup B = [0,2]$ and $A \cap B = \{1\} \neq \emptyset$ but clearly the outer measure of the union is the sum of the outer measure and both $A$ and $B$ will be measurable in the sense of Caratheodory (or in any sense, they are intervals! :D )
