Equicontinuity and convergence of measures

It is shown in this paper that each family of measures with values in an abelian topological group which is equicontinuous on a ring is equicontinuous on the generated -ring. A family of measures is equicontinuous iff the corresponding family of semivariations is equicontinuous. It is furthermore shown that a family of measures which is equicontinuous and Cauchy convergent on a ring is Cauchy convergent on the generated -ring. A family of measures which is Cauchy convergent for all countable sums of elements of a ring is Cauchy convergent on the generated -ring.