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.