V N U . JO U R N AL OF SCIENCE, M a th e m a tics - Physics. T . x x , N0 1 - 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.
1. 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) = 1 .

(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

X® respectively ( note that Ui can be taken from a fixed

basis of topology of X) .