I would like to find exemples to show and demonstrate that each of the statements of the definition of:
$\mu \left( \bigcup A_n\right)=\sum \mu \left( A_n\right)$ $\mu$ defined from a subset of partition of a given set to $\left[0;+\infty\right]$
are not reduntant.
Edit1: I mean Im looking for "applications" which can fit the finite additivity but not that associates the empty set to zero. Or the opposite.