A note on the absurd law of large numbers in economics

Let Γ be a Borel probability measure on R and (T, C, Q) a nonatomic probability space. Define H = {H ∈ C : Q(H) > 0}. In some economic models, the following condition is requested. There are a probability space (Ω,A, P ) and a real process X = {Xt : t ∈ T} satisfying for each H ∈ H, there is AH ∈ A with P (AH) = 1 such that t 7→ 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, C countably generated and Γ non trivial. 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. AMS 2010 Subject Classification. 60A05, 60A10, 28A05, 91B30, 91B70.