On nonatomic submeasures on N. II

It is shown that if two submeasures on N that are lim sup's of sequences of measures have the same zero sets, and one is nonatomic, so is the other, and they are (ε-δ)-equivalent. Moreover, if a submeasure η is the lim sup of a sequence of lower semi-continuous (lsc) submeasures and is 0-dominated by the so-called core γ• of an lsc submeasure γ, then η is also (ε-δ)-dominated by γ•. And if a submeasure η of this type is 0-dominated by a nonatomic submeasure, then it is nonatomic as well.