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- W4301900793 abstract "A tree $T$ is said to be homogeneous if it is uniquely rooted and there exists an integer $bgeq 2$, called the branching number of $T$, such that every $tin T$ has exactly $b$ immediate successors. We study the behavior of measurable events in probability spaces indexed by homogeneous trees. Precisely, we show that for every integer $bgeq 2$ and every integer $ngeq 1$ there exists an integer $q(b,n)$ with the following property. If $T$ is a homogeneous tree with branching number $b$ and ${A_t:tin T}$ is a family of measurable events in a probability space $(Omega,Sigma,mu)$ satisfying $mu(A_t)geqepsilon>0$ for every $tin T$, then for every $0<theta<epsilon$ there exists a strong subtree $S$ of $T$ of infinite height such that for every non-empty finite subset $F$ of $S$ of cardinality $n$ we have [ muBig(bigcap_{tin F} A_tBig) meg theta^{q(b,n)}. ] In fact, we can take $q(b,n)= big((2^b-1)^{2n-1}-1big)cdot(2^b-2)^{-1}$. A finite version of this result is also obtained." @default.
- W4301900793 created "2022-10-06" @default.
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- W4301900793 date "2011-05-12" @default.
- W4301900793 modified "2023-10-16" @default.
- W4301900793 title "Measurable events indexed by trees" @default.
- W4301900793 doi "https://doi.org/10.48550/arxiv.1105.2417" @default.
- W4301900793 hasPublicationYear "2011" @default.
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