Extension of measures on pseudo-D-lattices

Abstract

We prove a Carathéodory type extension theorem for

Extension of modular functions and measures

“…As a consequence, they obtained an extension theorem for measures which can be "controlled" by a family of positive measures; in particular, this implies (employing a control measure theorem) an extension theorem for measures defined on an orthomodular lattice with values in a locally convex space. These results of [2] are generalized in [3] for modular measures on D-lattices 2 and recently for modular measures on pseudo-D-lattices [4].…”

“…Here we give an essentially easier proof, which could be of interest also in the Boolean case. Such an extension theorem was proved in [4,Corollaries 5.5 and 5.6] under the additional assumption that G is a locally convex linear space, or that G is a normed group and μ has finite variation.…”

Pseudo-D-lattices and separating points of measures

“…6. The lattice L(R n ) of all linear subspaces of R n (where n > 1) is an infinite modular orthomodular lattice of finite height.…”

“…6. We say that a lattice L is dense in itself if, for every a, b ∈ L with a < b, there exists c ∈ L with a < c < b.…”

“…Example 7. 6. Let A be the algebra of finite and co-finite subsets of N, and denote by Z 2 the group of integers modulo 2.…”