I am trying to prove that Jordan measures satisfy with the following properties $A, B \subset \mathbb R$ and $d(A) = \sup( \{ d(x,y) | x,y \in A \} ) < \infty$, similarly for $B$:
$$\bar{\mu} (A) \leq \bar{\mu} (A \cup B) \leq \bar{\mu}(A) + \bar{\mu}(B)$$
I am uncertain of the terminology here. So I want to know why there is a restriction $d(A) = \sup( \{ d(x,y) | x,y \in A \} ) < \infty$? How does it change the problem without and with the assumption? What kind of things we have if we do not have the assumption about the limit?