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If $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space, is there a general definition for the "$k$-dependence" of an arbitrary family $(\mathcal{F}_i)_{i \in I}$ of sub-$\sigma$-algebra of $\mathcal{F}$ ?

Already know the definition for a sequence (if for all $n \in \mathbb{N},\sigma(\mathcal{F}_1,...,\mathcal{F}_n)$ and $\sigma(\mathcal{F}_{n+k+1},...)$ are independent) and that $k=0$ corresponds to the independence.

How can we define it for an arbitrary family? Is there a reference for this notion?

This question was asked here with no answers: https://math.stackexchange.com/questions/3989446/general-definition-for-k-dependence-of-a-family-of-sub-sigma-algebra

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    $\begingroup$ In the definition for the sequence case, $k$ represents a distance in the index set $\mathbb N$. To reasonably generalize to arbitrary index sets $I$, you need a notion of distance in $I$. $\endgroup$
    – Andreas Blass
    Commented Jan 19, 2021 at 17:52

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One possible definition is to assume a graph structure on the index set, where the maximal degree is $k$. Then you assume that every algebra $\mathcal{F}_i$ is independent of the join of the algebras $\{\mathcal{F}_j: j \; \text{not connected to} \; i\}$. Then you are in position to apply the Lovász local lemma.

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