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 σ-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 σ-ring. A family of measures which is Cauchy convergent for all countable sums of elements of a ring is Cauchy convergent on the generated σ-ring.
KeywordsNumber Theory Algebraic Geometry Topological Group Abelian Topological Group
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