On the representation of measures over bounded lattices

TL;DR

This paper explores how measures over bounded lattices can be represented using Boolean lattices and schemes, unifying previous approaches and extending the theoretical framework with algebraic geometry techniques.

Contribution

It introduces a unifying representation of measures over bounded lattices via Boolean lattices and schemes, expanding the theoretical understanding of lattice measures.

Findings

Measures over any bounded lattice can be represented as measures over a Boolean lattice.

Existence of a scheme associated with a bounded lattice such that measures correspond scheme-theoretically.

Characterization of measurability for finite 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.