Right closure of martingale sequences in the sense of the \(A\)-integral (Q5930774)

The usual notion of a martingale (with respect to Lebesgue integration) is generalized to that of a martingale in the sense of an \(A\)-integral. Sufficient conditions are given for such a martingale \(\{X_n,{\mathcal F}_n,\geq 1\}\) to be closed from the right by a random variable \(X_\infty\), along with conditions tantamount to closure from the right in a special case where, as \(C\to \infty\), \(CP[|X_n|> C]\to 0\) uniformly in \(n\) in a certain sense.