Measurable events indexed by products of trees

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 t ∈ T has exactly b immediate successors. A vector homogeneous tree T is a finite sequence (T1, ..., Td) of homogeneous trees and its level product ⊗T is the subset of the cartesian product T1 × ...× Td consisting of all finite sequences (t1, ..., td) 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.