By Lemma 27 the dual of [M.sup.p] is isometric to the space of countably additive measures on [beta]([K.sub.p]); therefore, for some v [member of] M([beta][K.sub.p])) with [parallel]v[parallel] = 1 and for all f [member of] [M.sup.p], one has

The previous inequality and Hahn-Banach theorem yield a norm-preserving extension of l to a continuous functional on C([beta]([K.sub.p])), which is a countably additive measure [??] on [beta]([K.sub.p]), such that [parallel][??][parallel] = 1 and [absolute value of <[??], f>] [less than or equal to] [tau](f) for every f [member of] C([beta]([K.sub.p])).

A Borel measure [mu] on a topological space X is purely finitely additive (p.f.a.) if whenever v is a nonnegative countably additive Borel measure on X bounded by [mu], in the sense that 0 [less than or equal to] v [less than or equal to] [absolute value of [mu]], then v = 0.

Throughout the paper m : [summation] [right arrow] X will be a positive countably additive vector measure, i.e m(A) [greater than or equal to] 0 for all A [member of] [summation].

It is clearly countably additive and then the corresponding space [L.sup.2](v) is well-defined.

The positive countably additive vector measure v(A) = [([[integral].sub.A][x.sup.2n]dx).sup.[infinity].sub.n=0] [member of] [c.sub.0] is then well defined.

If A is a [sigma]-algebra of subsets of a set Y, [mu]: A [right arrow] E a countably additive vector measure and p [member of] P, we denote the p-semi-variation of [mu] by [[bar.[mu]].sub.p], [[bar.[mu]].sub.p] (A) = sup{[absolute value of g x [mu]] (A): g [member of] [V.sup.0.sub.p]} (here [V.sup.0.sub.p] is the polar of [V.sub.p] in the duality <E,E'>) [8]; we consider the submeasure [[??].sub.p]: A [right arrow] [R.sup.+], [[??].sub.p](A) = sup{[[parallel][mu](B)[parallel].sub.p]: B [member of] A, B [subset] A} ([2], [5]).

Then it has a unique extension to a countably additive Borel measure [mu]: B(X) [right arrow] E which is inner regular by closed sets and outer regular by open sets.

Take any B [member of] B(X) and select any [??] [member of] A such that [??][intersection] X = B; define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; it is a trivial verification that [mu] is well-defined, is countably additive and it is inner regular by closed sets in X and outer regular by open sets in X.

Then [[mu].sub.n] is a

countably additive, regular complex measure with compact support contained in K(see the proof of Singer's theorem in (2)).

A real valued bounded additive set function is called countably additive if it assigns the countable sum of the values to a countable union of disjoint sets.

Probability measures on the real numbers, R, or on the integers Z, are typical examples of such countably additive functions.