A note on the absurd law of large numbers in economics

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