## 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 *t* ∈ *T* 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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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