Lattice Valuations: a Generalisation of Measure and Integral
Abraham Westerbaan · arXiv (Cornell University) · 2019
Measure and integral are two closely related, but distinct objects of study. Nonetheless, they are both real-valued lattice valuations: order preserving real-valued functions $ϕ$ on a lattice $L$ which are modular, i.e., $ϕ(x)+ϕ(y) = ϕ(x\wedge y)+ϕ(x\vee y)$ for all $x,y \in L$. We unify measure and integral by developing a theory for lattice valuations. We allow these lattice valuations to take their values from the reals, or any suitable ordered Abelian group.