Throughout the paper m : [summation] [right arrow] X will be a positive countably additive vector measure, i.

It is clearly countably additive and then the corresponding space [L.

It is well defined, and since the set is uniformly integrable, it is countably additive.

We define a countably additive vector measure n : [summation] [right arrow] [c.

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.

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.

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.

Let [mu] be a countably additive, regular, vector valued, Borel measure on R taking values in [B *.

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.

A real valued bounded additive set function [Phi] on (S, [Sigma]) is called purely finitely additive (see Yosida and Hewitt 1952) if whenever a countably additive function v satisfies: