Uniform absolute continuity in spaces of set functions

Let X be a regular topological space, K a collection of bounded regular measures defined on the Borel sets of X. The following conditions are equivalent. (1) Let M(X) denote the Borel measures, M(X) the nonnegative members of M(X). There is a A EM(X) + such that K is uniformly X-continuous. (2) If IUnIn = 1, 2,. .. I is a disjoint sequence of open sets, then limn-4Un) = 0 uniformly for ,u E K. (3) If E is a Borel subset of X and E > 0, there is a compact set F C E such that 1L|(E F) 0 we can find a compact set K C E such that Iy(E-K)I 0, there is a 8 > 0 such that if E is a Borel set with A(E) < 8, then tL(E)I < E for