Where does "Additivity" show up besides measure theory? The properties of "additivity" or "$\sigma$-additivity" seem to be quite localized phenomenons at first glance, specific to measures or appropiate generalizations of those. 
Let $L$ be a lattice. Elements $x,y\in L$ are disjoint, if $x\wedge y = \bot$ (smallest element). A map $f : L \to M$ into some commutative monoid $M$ is additive, if $f(x \vee y) = x + y$ whenever $x,y$ are disjoint. Something similar can be done with $\sigma$-additivity, where $M$ is a complete monoid.
I feel like there should be some interesting examples, where $L$ is not just a set of sets ordered by $\subseteq$. So:

Where does "Additivity" show up besides measure theory?

(A more basic problem is perhaps finding instances, where disjointness is useful. Here are two examples: $x,y\in \mathbb Z_{\geq 0}$ are disjoint w.r.t. to $\mid$, if they are coprime; subgroups $M,N\subseteq G$ are disjoint, if $M\cap N \cong 1$. In totally ordered sets, disjointness is of course quite boring).
 A: At some emotional level, the proof on page 17 of this paper$^{\,\large\dagger}$ is my favorite part of it with Section 4.2 on pages 17 through 22 following closely. This is about additivity-without addition on structures that are no lattices but have some things in common with them. The purpose is to explain why additivity of probabilities should apply when the probabilities are purely epistemic, so that one has no relative frequencies or proportions of populations, etc.
Whether this is "measure theory" may be something one could argue about.
$\dagger$ My paper "Scaled Boolean Algebras", Advances in Applied Mathematics, 2002. (A pdf file.)
A: Here's an example which is certainly not from measure theory:
Consider the lattice structure on the positive integers given by divisibility.  That is, $x \leq y$ if $x | y$.  Here the meets and joins are given by lcms and gcds, respectively, so "disjoint" means "relatively prime."  Then an additive (resp. multiplicative) function in the sense of number theory is precisely an additive function on this lattice with codomain the integers under addition (resp. multiplication).
