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 is a probability space ( Ω , A , P ) and a real process X = { X t : t ∈ T } satisfying for each H ∈ H , 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.