Let $W$ be a set, and let $v$ be a finitely additive probability measure on $2^W$.

Equip $2^W$ with the Borel sigma-algebra $\mathcal{B}$ generated by the sub-basic sets of the form $\{a: w \in a\}$ and $\{a: w \notin a\}$, where $w \in W$.

I am interested in characterizing those probability measures $P$ on $(2^W, \mathcal{B})$ such that for ($P$-almost) all $a \in 2^W$, $$P(\{w \in 2^W: v(w \cap a)=v(a)v(w)\})=1. \tag{1}$$ Intuitively, this is saying that for all $a$, almost every $P$-randomly selected subset $w$ of $W$ is independent of $a$.

One result along these lines comes from a well-known paper of Fremlin and Talagrand. In particular, they show that if $v(a)=0$ for all finite $a \subset W$ and $v$ is measurable with respect to $P$ defined by $$P(\{a: b \subset a\}) = 2^{-|b|} \ \text{for all finite } b \subset W, \tag{2}$$ then (1) holds for all $a \in 2^W$ (see Lemma 1B and 1J).

Are there generalizations of the Fremlin-Talagrand framework that consider $P$ other than the "iid fair coin" defined in (2)? How can we relax the assumption that $v$ vanishes on finite sets? Any more recent references are appreciated.

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