The Non-closeness and Non-denseness of H~∞ in B.MO

The purpose of this paper is to oite an example.We construot a referencesystem(Ω0,0,(0)(t≥0),p0),on which the martingale space ~∞ is not closed andnot dense in the martingale space .To find a suitable closed sub-space or a suitable dense sub-space for the ,especially the later is valuable.It has been proved in [1] (Theorem 10) that,if ~∞