Range of density measures

summary:We investigate some properties of density measures – finitely additive measures on the set of natural numbers $\text{$\mathbb {N}$}$ extending asymptotic density. We introduce a class of density measures, which is defined using cluster points of the sequence $\bigl (\frac{A(n)}{n}\bigr )$ as well as cluster points of some other similar sequences. We obtain range of possible values of density measures for any subset of $\text{$\mathbb {N}$}$. Our description of this range simplifies the description of Bhashkara Rao and Bhashkara Rao [Bhaskara Rao, K. P. S., Bhaskara Rao, M., Theory of Charges – A Study of Finitely Additive Measures, Academic Press, London–New York, 1983.] for general finitely additive measures. Also the values which can be attained by the measures defined in the first part of the paper are studied