Measurable events indexed by products of trees

Abstract

A tree T is said to be homogeneous if it is uniquely rooted and there exists an integer b ≥ 2, called the branching number of T, such that every tT has exactly b immediate successors. A vector homogeneous tree T is a finite sequence (T 1,...,T d ) of homogeneous trees and its level product ⊗T is the subset of the Cartesian product T 1×...×T d consisting of all finite sequences (t 1,...,t d ) of nodes having common length.

We study the behavior of measurable events in probability spaces indexed by the level product ⊗T of a vector homogeneous tree T. We show that, by refining the index set to the level product ⊗S of a vector strong subtree S of T, such families of events become highly correlated. An analogue of Lebesgue’s density Theorem is also established which can be considered as the “probabilistic” version of the density Halpern-Läuchli Theorem.

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Correspondence to Pandelis Dodos.

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Dodos, P., Kanellopoulos, V. & Tyros, K. Measurable events indexed by products of trees. Combinatorica 34, 427–470 (2014). https://doi.org/10.1007/s00493-014-2880-2

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Mathematics Subject Classification (2000)

  • 05D10
  • 05C05