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 = {Xt : t ∈ T } satisfying for each H ∈H, there is AH ∈ A with P(AH ) = 1 such that t → 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 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
Patrizia Berti, M.G. (2012). A note on the absurd law of large numbers in economics. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 388(1), 98-101 [10.1016/j.jmaa.2011.10.040].
A note on the absurd law of large numbers in economics
Pietro Rigo
2012
Abstract
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 = {Xt : t ∈ T } satisfying for each H ∈H, there is AH ∈ A with P(AH ) = 1 such that t → 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 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 distributionI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
