A note on the law of large numbers in economics

Let $(S,\mathcal{B},\Gamma)$ and $(T,\mathcal{C},Q)$ be probability spaces, with $Q$ nonatomic, and $\mathcal{H}=\{H\in\mathcal{C}:Q(H)>0\}$. In some economic models, the following conditional law of large numbers (LLN) is requested. There are a probability space $(\Omega,\mathcal{A},P)$ and a process $X=\{X_t:t\in T\}$, with state space $(S,\mathcal{B})$, satisfying \begin{gather*} \text{for each }H\in\mathcal{H},\text{ there is }A_H\in\mathcal{A}\text{ with }P(A_H)=1\text{ such that } \\t\mapsto X(t,\omega)\text{ is measurable and }\,Q\bigl(\{t:X(t,\omega)\in\cdot\}\mid H\bigr)=\Gamma(\cdot)\,\text{ for }\omega\in A_H. \end{gather*} If $\Gamma$ is not trivial and the $\sigma$-field $\mathcal{C}$ 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 $\mathcal{P}$ of finite dimensional distributions, there are a finitely additive probability space $(\Omega,\mathcal{A},P)$ and a process $X$ such that $X\sim\mathcal{P}$ and the conditional LLN is satisfied.