A theorem on martingale selection for relatively open convex set-valued random sequences

For set-valued random sequences (G _n) _n=0 ^N with relatively open convex values G _n(ω), we prove a new test for the existence of a sequence (x _n) _n=0 ^N of selectors adapted to the filtration and admitting an equivalent martingale measure. The statement is formulated in terms of the supports of regular upper conditional distributions of G _n. This is a strengthening of the main result proved in our previous paper [1], where the openness of the set G _n(ω) was assumed and a possible weakening of this condition was discussed.