manuscripta math. 5t 123 - 131 (I97l) О by Springer-Verlag 1971

EQUIGONTINUITY AND CONVERGENCE OF MEASURES

Dieter Landers and Lothar Rogge

It is shown in this paper that each family of measures v/ith values in an abelian topological group which is continuous on a ring is equicontinuous on the generated cT-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 C-ring. A family of measures which is Cauchy convergent for all countable sums of ments of a ring is Cauchy convergent on the generated cr-ring.

1 . Preliminaries

In this paper a topological group G is always assumed to

be abelian. The system of neighborhoods of the zero ment of G is denoted by 1C(0). A sequence a eG,n€ IN, is

Cauchy convergent iff it is Cauchy convergent with respect

to the uniformity

{ { ( a , b ) 6 G X G : а-Ьб U } : U € U(0)} .

Let TR be a CT-ring on X; a functionyu :'S?*—>G is a measure

iff for all sequences of disjoint sets A.cTfl ,i£(N, the

sequence ( £ZI M-C^-i ))r.^n\r converges to >u.( 5 A. ). The i=1^ 1 neiLM ^^^ 1

system of all measures ju, :lR,->Gr is denoted by M(1R ,G).

Let m be a subsystem of 'R and A€ 'R . We write

1R . J\A := {ВлА : Bt ^^] and denote by Ifi^ the system of

all countable unions of elements of 'R . If R is a rine:, cr о о '^'^

Ж is the system of all countable sums of elements of ^ ,

IfWcM ( 'R , G ) , ^C^ , we write ^(^):={/t(S):yLueWï, S€^}cG.^

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