Measure free martingales and martingale measures (Q2655532)

Let \(\Omega\neq\emptyset\) be any set and let \(\emptyset\neq T\in[0,\infty)\) be countable. Let \(f_t\) \((t\in T)\) be a family of real-valued functions on \(\Omega\), and let \(Q_t:=\{f^{-1}_t(a): a\in f_t(\Omega)\}\) denote the partition of \(\Omega\), generated by \(f_t\). For elements \(t_1<\cdots< t_k\) of \(T\) let \(Q_{t_1,\dots,t_k}\) denote the partition generated by \(Q_{t_1,\dots,t_k}\) and let \(f_{t_k}\equiv: f_{t_k}(q)\) on \(q\in Q_{t_1,\dots,t_k}\). The family \((f_t)\) is called a measure free martingale if for any subset \(t_1<\dots< t_{k+1}\) of \(T\) and any \(q\in Q_{t_1,\dots,t_k}\), the value \(f_{t_k}(q)\) belongs to the convex hull of the values assumed by \(f_{t_{k+1}}\) on \(q\). The main result of the present paper is the following Theorem. Let \((f_t)\) \((t\in T)\) be a measure-free martingale on \(\Omega\) such that \(T\) is countable, each \(f_t\) is bounded, and \((f_t)\) separates the points of \(\Omega\). Then there exists a compact metric space \(\Omega'\) in which \(\Omega\) is densely embedded, and there exist continuous functions \(f'_t\) \((t\in T)\) on \(\Omega'\) such that (i) for all \(t\in T\), \(f_t'\) extends \(f_t\); (ii) \((f'_r)\) is a measure free martingale, and (iii) there exists a probability measure \(\mu\) on the Borel subsets of \(\Omega'\) with respect to which \((f'_t)\) is a martingale, i.e., for any \(t_1<\cdots< t_k\) \(_\mu[f_{t_k}'\mid f_{t_1}',\dots,f_{t_{k-1}}]=f_{t_{k-1}}\) a. e. It is easy to see (using regular conditional probabilities) that when \(T\) is countable and \((f_t)\) \((t\in T)\) is a martingale defined on a probability space \((X,{\mathcal B}, \mu)\), then there exists a \(\mu\)-null set \(N\) such that \((f_t)\) is a measure free martingale on \(X\setminus N\). The authors prove a sort of converse of this statement.