DSpace at VNU: Probability measure functors preserving the regular property

V N U JO U R N AL OF SCIENCE, M a th e m a tics - Physics T x x , N0 - 2004 P R O B A B IL IT Y M E A SU R E FUNCTORS PR E SE R V IN G THE REG ULAR PRO PERTY Ta K hac Cu Department o f Mathematics, Vinh University A bstract Let X be a topological Hausdorff space For each k E N, by Pk ( X) we denote the set of all probability measures on X , whose supports of no more than k points Then probability measure functor Pk preserve the regular property Probability m ea su r e w ith finite su p p o rts Let X be a topological Hausdorff space A probability measure with finite supports on X is a function Ị1 : X —>• [0,1] satisfying the condition supp/i = {x G X : /i(x) > 0} (a) is finite fl(x) = (b) xGsupp/i For each k G N, let Pk{X) denote the set of all probability measure on X whose supports of no more than k points Then every /i G Pk{X) can be written in the form Q where ỏj: is Dirac function, th a t is y +x y =x and Q TUi = n (X i) > , TUj = G Pk( X) 1=1 T y p e se t by 19 Ta Khac Cu 20 has a neighborhood of the form 0(ụ.Q,U\)U2,—,Uq,è), where e > 0; U\ , Ư2 , disjoint neighborhood of ^ 2, Uq are X® respectively ( note that Ui can be taken from a fixed basis of topology of X)