On the representation of measurable functions by martingales (Q802204)

Let (\(\Omega\),\({\mathcal F},P)\) be a complete probability space. A d-sequence is a filtration \({\mathcal F}_ n\uparrow {\mathcal F}\) with the property that for every n if \(A\in {\mathcal F}_ n\) and \({\mathcal F}_{n| A}={\mathcal F}_{| A}\), then \(P(A)=0\). \textit{C. W. Lamb} proved [Trans. Am. Math. Soc. 188, 395-405 (1974; Zbl 0283.60052)] that if (\({\mathcal F}_ n)_ n\) is a d-sequence, then every \({\mathcal F}\)-measurable a.s. finite random variable f is the a.s. limit of some \({\mathcal F}_ n\)-martingale. The author extends this result in several directions and proves also a converse: if every extended random variable (i.e. \(f: \Omega\) \(\to \bar R)\) is the limit in probability of some \({\mathcal F}_ n\)-martingale then \({\mathcal F}_ n\) is a d-sequence. It is a very interesting and well written paper which suggest further questions: for instance what can be said about a filtration \({\mathcal F}_ n\uparrow {\mathcal F}\) with the property that every r.v. is the a.s. limit of a martingale? It seems that the answer must be in terms of some metric on the set of all sub-algebras on \({\mathcal F}\). The intermediary results of this paper are also of interest.