Density measures and additive property

We consider a certain class of normalized positive linear functionals on l∞ which extend the Cesàro mean. We study the set of its extreme points and it turns out to be the set of linear functionals constructed from free ultrafilters on natural numbers N. Also, regarding them as finitely additive measures defined on all subsets of N, which are often called density measures, we study a certain additivity property of such measures being equivalent to the completeness of the Lp-spaces on such measures. Particularly a necessary and sufficient condition for such a density measure to have this property is obtained.