The Range of a Module Measure Defined on an Effect Algebra

Abstract

Effect algebras are the main object of study in quantum mechanics. Module measures are those measures defined on an effect algebra with values on a topological module. Let R be a topological ring and M a topological R-module. Let L be an effect algebra. The range of a module measure μ:L→M is studied. Among other results, we prove that if L is an sRDP σ-effect algebra with a natural basis and μ:L→R is a countably additive measure, then μ has bounded variation.

On Focal Borel Probability Measures

“…A classical measure theory result establishes that the measure of the union a countable increasing family of measurable subsets can be computed as the limit of the sequence of the measures of the subsets. This result was transported in [8] to the scope of measures defined on a effect algebra and valued on a topological module over a topological ring. Here, we extend [8] to uncountable families with countable cofinal subsets.…”

“…This result was transported in [8] to the scope of measures defined on a effect algebra and valued on a topological module over a topological ring. Here, we extend [8] to uncountable families with countable cofinal subsets. However, we first recall [8] and prove it for the sake of completeness.…”

“…Here, we extend [8] to uncountable families with countable cofinal subsets. However, we first recall [8] and prove it for the sake of completeness.…”