A note on the law of large numbers in economics - Patrizia Berti; Michele Gori; Pietro Rigo (Università degli Studi di Pavia, Dipartimento di Economia Politica e Metodi Quantitativi (EPMQ), 2010)
Let (S, B, G ) and (T, C,Q) be probability spaces, with Q nonatomic, and H = h in C : Q(H) > 0. In some economic models, the following conditional law of large numbers (LLN) is requested. There are a probability space (O,A,P) and a process X = Xt : t in T, with state space (S, B), satisfying for each h in H, there is Ah in A with P(Ah) = 1 such that t –> X(t, w) is measurable and Q t : X(t, w) in · | h = G(·) for w in Ah. If is not trivial and the sigma-field C countably generated, the conditional LLN fails in the usual (countably additive) setting. Instead, as shown in this note, it holds in a finitely additive setting. Also, X can be taken to have any given distribution. In fact, for any consistent set P of finite dimensional distributions, there are a finitely additive probability space (O,A,P) and a process X such that X tilde P and the conditional LLN is satisfied.
Keywords: 330; Aggregate uncertainty; Extension; Finitely additive probability measure; Individual risk; Law of large numbers