Existence theorems for measures on continous posets, with applications to random set theory.

Abstract

We state conditions on a partially ordered set L and a mapping λ, defined on a class of filters on L, under which λ extends to a measure on the minimal σ-field over this class. By applying this result to the case when L is a continuous lattice, all locally finite measures on L are identified as well as all Levy-Khinchin measures. We then characterize these kinds of measures on continuous semilattices and continuous posets. The correspondence between probability measures on the line and distribution function is a particular case of this result. We give a simple proof of the Daniell-Kolmogorov existence theorem for probability measures on products of continuous lattices