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I want to consider a variant on conglomerability, which I will call independence conglomerability.
Suppose we have a collection of events {Ei : i ∈ I}, and suppose that J is a randomly chosen member of I, with J independent of all the Ei taken together.
Independence conglomerability requires that if P(Ei) ≥ λ for all i, then P(EJ) ≥ λ, where ω ∈ EJ if and only if ω ∈ EJ(ω) for ω in our underlying probability space Ω.