Orthogonality and Disjointness in Spaces of Measures

TL;DR

This paper explores the geometric and measure-theoretic structures of measure cones in probabilistic theories, introducing orthogonality and disjointness concepts to interpret measure decompositions and their implications.

Contribution

It introduces a geometric interpretation of measure disjointness and orthogonality within measure cones, linking these to measure decompositions and extending measure-theoretic understanding.

Findings

Characterizes measure disjointness via orthogonality in measure cones

Provides a geometric interpretation of the Hahn-Jordan decomposition

Suggests measure cones admit measure-theoretic interpretations

Abstract

The convex and metric structures underlying probabilistic physical theories are generally described in terms of base normed vector spaces. According to a recent proposal, the purely geometrical features of these spaces are appropriately represented in terms of the notion of `measure cone' and the `mixing distance' [1], a specification of the novel concept of `direction distance' [2]. It turns out that the base norm is one member of a whole characteristic family of `mc-norms' from which it can be singled out by virtue of a certain orthogonality relation. The latter is seen to be closely related to the concept of minimal decomposition. These connections suggest a simple geometric interpretation of the familiar notion of the disjointness of (probability) measures and the Hahn-Jordan decomposition of measures which has been addressed briefly in [1] and will be elaborated here. The results obtained give an indication of the extent to which a general measure cone admits measure theoretic interpretations. [1] P. Busch, E. Ruch: The Measure Cone -- Irreversibility as a Geometrical Phenomenon, Int. J. Quant. Chem. 41 (1992) 163-185. [2] E. Ruch: Der Richtungsabstand}, Acta Applic. Math. 30 (1992) 67-93.