Factorizing random sets and type III Arveson systems

TL;DR

This paper develops a measure-theoretic framework for constructing Arveson systems from random sets, introduces a method for creating type III systems via infinite products, and applies it to Brownian motion zero sets.

Contribution

It provides a new measure-theoretic characterization of Arveson systems and introduces a general construction method for type III systems from type II$_0$ seeds.

Findings

Canonical generation of Arveson systems from measurable families of measures.

Characterization of spatiality via measure factorization.

Explicit construction of type III systems from Brownian motion zero sets.

Abstract

We develop a representative-level framework for the Liebscher-Tsirelson random-set construction of Arveson systems from stationary factorizing measure types. We introduce the notion of a measurable factorizing family of probability measures on hyperspaces of closed subsets of time intervals and prove that every such family canonically generates an Arveson system. Within this framework we obtain a purely measure-theoretic characterization of spatiality: positive normalized units correspond exactly to dominated families of measures that factorize strictly. We then present a general mechanism for constructing type III Arveson systems via infinite products of measurable factorizing families. Starting from a type II$_0$ seed satisfying a quantitative Hellinger-smallness condition, we form a marked infinite product indexed by $[0,1]\times\mathbb N$ and show, using Kakutani's criterion, that the resulting product system admits no units. This yields a robust construction principle for type III random-set systems. As an application we analyze zero sets of Brownian motion. After anchor-adapted localization and Palm-type uniformization, the Brownian seed satisfies the required overlap estimates, and the associated infinite-product construction produces explicit examples of type III random-set systems, as anticipated in the work of Tsirelson and Liebscher.