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 $b\meg 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. A vector homogeneous tree $\mathbf{T}$ is a finite sequence $(T_1,...,T_d)$ of homogeneous trees and its level product $\otimes\mathbf{T}$ is the subset of the cartesian product $T_1\times ...\times T_d$ 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 $\otimes\mathbf{T}$ of a vector homogeneous tree $\mathbf{T}$. We show that, by refining the index set to the level product $\otimes\mathbf{S}$ of a vector strong subtree $\bfcs$ of $\mathbf{S}$, 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--L\"{a}uchli Theorem.