On the representation of measures over bounded lattices

Abstract

In this paper we investigate measures over bounded lattices, extending and giving a unifying treatment to previous works. In particular, we prove that the measures of an arbitrary bounded lattice can be represented as measures over a suitably chosen Boolean lattice. Using techniques from algebraic geometry, we also prove that given a bounded lattice $X$ there exists a scheme $\mathcal{X}$ such that a measure over $X$ is the same as a (scheme-theoretic) measure over $\mathcal{X}$. We also define the measurability of a lattice, and describe measures over finite lattices.