Let E be a Banach space of measurable functions, and X a Banach space. It is known that Eis dense in the space E(X) of vector-valued functions if the condition (A) holds:The necessity of this condition was shown in the paper cited in the title (MR53 #8920) under the assumption that X contains a complemented infinite-dimensional subspace with unconditional basis. In the present paper the requirement of the existence of such a subspace is removed. Also, an error in the earlier paper is corrected.Bibliography: 7 titles.