Factorizing random sets and type III Arveson systems

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.