A note on the absurd law of large numbers in economics

Let Γ be a Borel probability measure on the set of real numbers R and (T,C,Q) a nonatomic probability space. In some economic models, the following condition is requested. There is a probability space (Ω,A,P) and a real process X = {X_t : t ∈ T } satisfying the following condition: for each H ∈ C with Q (H) > 0, there is A_H ∈ A with P(A_H ) = 1 such that t → X(t,ω) is measurable and Q ({t: X(t,ω) ∈ ·}|H)=Γ(·) for ω ∈ A_H . Such a condition fails if P is countably additive, C countably generated and Γ nontrivial. Instead, as shown in this note, it holds for any C and Γ under a finitely additive probability P. Also, X can be taken to have any given distribution.