More on measurable algebras and Rademacher systems with applications to analysis of Riesz spaces

TL;DR

This paper characterizes when certain families in Boolean algebras induce unique measures, introduces Rademacher systems in Riesz spaces, and develops a new measure-based integration theory with applications to analysis.

Contribution

It establishes necessary and sufficient conditions for complete Rademacher families, links these to measure and homogeneity in Boolean algebras, and extends Rademacher and Haar systems to Riesz spaces with a new integration framework.

Findings

Complete Rademacher families induce unique measures on Boolean algebras.

Boolean algebras with such systems are isomorphic if they have the same cardinality.

A new measure-based integration theory is developed for Riesz spaces.

Abstract

We find necessary and sufficient conditions on a family $\mathcal{R} = (r_i)_{i \in I}$ in a Boolean algebra $\mathcal{B}$ under which there exists a unique positive probability measure $\mu$ on $\mathcal{B}$ such that $\mu ( \bigcap_{k=1}^n \theta_k r_{i_k} ) = 2^{-n}$ for all finite collections of distinct indices $i_1, \ldots, i_n \in I$ and all collections of signs $\theta_1, \ldots, \theta_n \in \{-1,1\}$, where the product $\theta x$ of a sign $\theta$ by an element $x \in \mathcal{B}$ is defined by setting $1 x = x$ and $-1 x = - x = \mathbf{1} \setminus x$. Such a family we call a complete Rademacher family. We prove that Dedekind $\sigma$-complete Boolean algebras admitting complete Rademacher systems of the same cardinality are isomorphic. As a consequence, we obtain that a Dedekind $\sigma$-complete Boolean algebra is homogeneous measurable if and only if it admits a complete Rademacher family. This new way to define a measure on a Boolean algebra allows us to define classical systems on an arbitrary Riesz space, such as Rademacher and Haar. We define a complete Rademacher system of any cardinality and a countable complete Haar system on an element $e > 0$ of a vector lattice $E$ in such a way that if $e$ is an order unit of $E$ then the corresponding systems become complete for the entire $E$. We prove that if $E$ is Dedekind complete then any complete Haar system on $e$ is an order Schauder basis for the ideal $A_e$ generated by $e$. Finally, we develop a theory of integration in a Riesz space of elements of the band $B_e$ generated by a fixed $e > 0$ with respect to the measure on the Boolean algebra $\mathfrak{F}_e$ of fragments of $e$ generated by a complete Rademacher family on $\mathfrak{F}_e$. Much space is devoted to examples showing that our way of thinking is sharp (e.g., we show the essentiality of each of the condition in the definition of a Rademacher family).