asymptotic density, weighted asymptotic density

Let us imagine the set of natural numbers and the set of even numbers. From the point of cardinality, these two sets are the ``same'', i.e. have the same cardinality $\aleph_0,$ the cardinality of infinite countable set. But intuitively, one can say that even numbers are only half of the na\-tu\-ral numbers -- the probability with which we pick out an even number is $1/2$ and the same is for the odd numbers. So what is the mathematical tool that can reflect our intuition? The conventional name of this measures is ``density" with an adjective. If we want to consider the measure of greatness, which could divide the sets of positive integers from a~certain aspect of their structure, it is more convenient to consider the finitely additive measure. We shall study most known type -- so called asymptotic density.