In measure theory one usually starts with a $\sigma$-algebra $A$ of sets and considers a measure $\mu:A\to [0,\infty]$. I'm interested in abstracting this definition to allow more general domains and codomains for the measure function. I'm aware of measures allowed to take values in $\mathbb {R}$ as well as in $\mathbb {C}$ and I'm also aware of vector valued measures. I'm not really familiar with measures having a domain that is anything but a $\sigma$-algebra of sets (except of course for finitely additive measures).
So, any information or reference to work done along these lines would be greatly appreciated. Especially, for the most general case where the domain of $\mu$ is allowed to be any $\sigma$-complete lattice and the codomain is the most general kind of lattice (probably complete with some binary operation $+$) that will support a good theory. But also non so far-reaching generalizations would be great. Thanks.